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@ -1156,17 +1156,18 @@ For Stewart platform:
* Introduction :ignore:
- In the detail design phase, one goal is to optimize the design of the nano-hexapod
- Parts are usually optimized using Finite Element Models that are used to estimate the static and dynamical properties of parts
- However, it is important to see how to dynamics of each part combines with the nano-hexapod and with the micro-station.
One option would be to use a FEM of the complete NASS, but that would be very complex and it would be difficult to perform simulations of experiments with real time control implemented.
- The idea is therefore to combine FEM with the multi body model of the NASS.
To do so, Reduced Order Flexible Bodies are used (Section ref:sec:detail_fem_super_element)
- The theory is described
- The method is validated using experimental measurements
- Two main elements of the nano-hexapod are then optimized:
- The actuator (Section ref:sec:detail_fem_actuator)
- The flexible joints (Section ref:sec:detail_fem_joint)
During the detailed design phase of the nano-hexapod, optimizing individual components while ensuring their dynamic compatibility with the complete system presents significant challenges.
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.
A full finite element model of the assembled system, 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 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 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 amplified piezoelectric 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 precise control of directional 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
<<sec:detail_fem_super_element>>
@ -3373,6 +3374,18 @@ exportFig('figs/detail_fem_joints_fem_vs_perfect_iff_plant.pdf', 'width', 'half'
:END:
<<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 phase of the nano-hexapod.
The approach was first validated using an amplified piezoelectric actuator, where predicted dynamics showed excellent agreement with experimental measurements for both open and closed-loop behavior.
This validation established confidence in the method's ability to accurately predict component behavior within the broader system context.
The methodology was then successfully applied to optimize two critical components.
For the actuators, it enabled validation of the APA300ML selection while providing both high-fidelity and computationally efficient models for system simulation.
Similarly, for the flexible joints, the analysis of bending and axial stiffness effects led to clear specifications and an optimized design that balances competing mechanical requirements.
In both cases, the ability to seamlessly integrate finite element models into the multi-body framework proved essential for understanding component interactions and their impact on system-level dynamics.
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.
* Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

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% Created 2025-02-26 Wed 23:11
% Created 2025-02-27 Thu 01:24
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -24,26 +24,21 @@
\clearpage
\begin{itemize}
\item In the detail design phase, one goal is to optimize the design of the nano-hexapod
\item Parts are usually optimized using Finite Element Models that are used to estimate the static and dynamical properties of parts
\item However, it is important to see how to dynamics of each part combines with the nano-hexapod and with the micro-station.
One option would be to use a FEM of the complete NASS, but that would be very complex and it would be difficult to perform simulations of experiments with real time control implemented.
\item The idea is therefore to combine FEM with the multi body model of the NASS.
To do so, Reduced Order Flexible Bodies are used (Section \ref{sec:detail_fem_super_element})
\begin{itemize}
\item The theory is described
\item The method is validated using experimental measurements
\end{itemize}
\item Two main elements of the nano-hexapod are then optimized:
\begin{itemize}
\item The actuator (Section \ref{sec:detail_fem_actuator})
\item The flexible joints (Section \ref{sec:detail_fem_joint})
\end{itemize}
\end{itemize}
During the detailed design phase of the nano-hexapod, optimizing individual components while ensuring their dynamic compatibility with the complete system presents significant challenges.
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.
A full finite element model of the assembled system, 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 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 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 amplified piezoelectric 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 precise control of directional 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}
\label{sec:orgeb21f99}
\label{sec:orgfd22661}
\label{sec:detail_fem_super_element}
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.
@ -55,7 +50,7 @@ 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}).
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}
\label{sec:org494af25}
\label{sec:org93ab665}
\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.
@ -79,7 +74,7 @@ m = 6 \times n + p
\end{equation}
\section{Example with an Amplified Piezoelectric Actuator}
\label{sec:orgad4f3ec}
\label{sec:org1e66a5f}
\label{ssec:detail_fem_super_element_example}
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}.
@ -110,7 +105,7 @@ Stiffness & \(21\,N/\mu m\)\\
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage}
\paragraph{Finite Element Model}
\label{sec:orgb4da286}
\label{sec:orgce5afdb}
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 finite element mesh, shown in Figure \ref{fig:detail_fem_apa95ml_mesh}, was then generated.
@ -151,7 +146,7 @@ The modal reduction procedure was then executed, yielding the reduced mass and s
\end{figure}
\paragraph{Super Element in the Multi-Body Model}
\label{sec:org2f8920b}
\label{sec:org809b5e7}
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.
@ -163,7 +158,7 @@ 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.
\paragraph{Sensor and Actuator ``constants''}
\label{sec:org13dbb26}
\label{sec:orgb3a075f}
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}.
@ -222,7 +217,7 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
\end{table}
\paragraph{Identification of the APA Characteristics}
\label{sec:org83f2d6f}
\label{sec:orge041867}
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.
@ -248,7 +243,7 @@ Through the established amplification factor of 1.5, this translates to a predic
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}
\label{sec:org92e845e}
\label{sec:org354cea4}
\label{ssec:detail_fem_super_element_validation}
Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation.
The goal is to measure the dynamics of the APA95ML and compared it with predictions derived from the multi-body model incorporating the actuator as a flexible element.
@ -274,7 +269,7 @@ Measurement of the sensor stack voltage \(V_s\) was performed using an analog-to
\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}
\paragraph{Comparison of the dynamics}
\label{sec:orgfabbfd3}
\label{sec:orgc0ac08b}
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}.
@ -309,7 +304,7 @@ Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could
\end{figure}
\paragraph{Integral Force Feedback with APA}
\label{sec:orgd052e81}
\label{sec:org4b65d74}
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.
@ -343,7 +338,7 @@ The close agreement between experimental measurements and theoretical prediction
\end{figure}
\section*{Conclusion}
\label{sec:org0aca0da}
\label{sec:org105aef7}
The modeling procedure presented in this section will demonstrate significant utility for the optimization of complex mechanical components within multi-body systems, particularly in the design of actuators (Section \ref{sec:detail_fem_actuator}) and flexible joints (Section \ref{sec:detail_fem_joint}).
Through experimental validation using an Amplified Piezoelectric Actuator, the methodology has been shown to accurately predict both open-loop and closed-loop dynamic behavior, thereby establishing its reliability for component design and system analysis.
@ -353,7 +348,7 @@ This is exemplified by the nano-hexapod configuration, where the implementation
However, the methodology remains valuable for system analysis, as the extraction of frequency domain characteristics can be efficiently performed even with such high-order models.
\chapter{Actuator Selection}
\label{sec:orgb6d5574}
\label{sec:org3ec4809}
\label{sec:detail_fem_actuator}
The selection and modeling of actuators constitutes a critical step in the development of the nano-hexapod.
This chapter presents the approach to actuator selection and modeling.
@ -361,7 +356,7 @@ First, specifications for the nano-hexapod actuators are derived from previous a
Then, the chosen actuator is modeled using the reduced-order flexible body approach developed in the previous section, enabling validation of this selection 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}
\label{sec:org021ab2f}
\label{sec:org929a34b}
\label{ssec:detail_fem_actuator_specifications}
The actuator selection process was driven by several critical requirements derived from previous dynamic analyses.
@ -436,7 +431,7 @@ Height \(< 50\, [mm]\) & 22 & 30 & 24 & 27 & 16\\
\end{table}
\section{APA300ML - Reduced Order Flexible Body}
\label{sec:orgbc872e3}
\label{sec:orgeff9b1b}
\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}.
@ -464,7 +459,7 @@ While this high order provides excellent accuracy for validation purposes, it pr
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}
\label{sec:org798e5bf}
\label{sec:org120c274}
\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 was developed, adapted from \cite{souleille18_concep_activ_mount_space_applic}.
@ -537,7 +532,7 @@ While higher-order modes and non-axial flexibility are not captured, the model a
\end{figure}
\section{Electrical characteristics of the APA}
\label{sec:org67d72f6}
\label{sec:orga7af5a1}
\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}.
@ -558,7 +553,7 @@ Proper consideration must be given to voltage amplifier specifications and force
These aspects, being fundamental to system implementation, will be addressed in the instrumentation chapter.
\section{Validation with the Nano-Hexapod}
\label{sec:orgd7e1728}
\label{sec:orgba951c9}
\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.
@ -591,54 +586,12 @@ These results validate both the selection of the APA300ML and the effectiveness
\end{figure}
\chapter{Flexible Joint Design}
\label{sec:orgce87a36}
\label{sec:org93c9b2c}
\label{sec:detail_fem_joint}
The flexible joints have few advantages compared to conventional joints such as the \textbf{absence of wear, friction and backlash} which allows extremely high-precision (predictable) motion.
The parasitic bending and torsional stiffness of these joints usually induce some \textbf{limitation on the control performance}. \cite{mcinroy02_model_desig_flexur_joint_stewar}
In this document is studied the effect of the mechanical behavior of the flexible joints that are located the extremities of each nano-hexapod's legs.
Ideally, we want the x and y rotations to be free and all the translations to be blocked.
However, this is never the case and be have to consider:
\begin{itemize}
\item Non-null bending stiffnesses
\item Non-null radial compliance
\item Axial stiffness in the direction of the legs
\end{itemize}
This may impose some limitations, also, the goal is to specify the required joints stiffnesses.
Say that for simplicity (reduced number of parts, etc.), we consider the same joints for the fixed based and the top platform.
\textbf{Outline}:
\begin{itemize}
\item Perfect flexible joint
\item Imperfection of the flexible joint: Model
\item Study of the effect of limited stiffness in constrain directions and non-null stiffness in other directions
\item Obtained Specification
\item Design optimisation (FEM)
\item Implementation of flexible elements in the Simscape model: close to simplified model
\end{itemize}
\section{Flexible joints for Stewart platforms}
\label{sec:orgebc6043}
Review of different types of flexible joints for Stewart plaftorms (see Figure \ref{fig:detail_fem_joints_examples}).
Typical specifications:
\begin{itemize}
\item Bending stroke (i.e. long life time by staying away from yield stress, even at maximum deflection/load)
\item Axial stiffness
\item Bending stiffness
\item Maximum axial load
\item Well defined rotational axes
\end{itemize}
Typical values?
\begin{itemize}
\item \(K_{\theta, \phi} = 15\,[Nm/rad]\) stiffness in flexion
\item \(K_{\psi} = 20\,[Nm/rad]\) stiffness in torsion
\item \[ K_a = 60\,[N/\mu m] \] axial stiffness
\end{itemize}
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.
For Stewart platforms requiring nanometric precision, numerous flexible joint designs have been developed and successfully implemented, as illustrated in Figure \ref{fig:detail_fem_joints_examples}.
For design simplicity and component standardization, identical joints are employed at both ends of the nano-hexapod struts.
\begin{figure}[htbp]
\begin{subfigure}{0.35\textwidth}
@ -662,41 +615,36 @@ Typical values?
\caption{\label{fig:detail_fem_joints_examples}Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_yang}) \cite{yang19_dynam_model_decoup_contr_flexib}. (\subref{fig:detail_fem_joints_preumont}) \cite{preumont07_six_axis_singl_stage_activ}. (\subref{fig:detail_fem_joints_wire}) \cite{du14_piezo_actuat_high_precis_flexib}.}
\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}.
This section examines how these non-ideal characteristics affect system behavior, focusing particularly on bending/torsional stiffness (Section \ref{ssec:detail_fem_joint_bending}) and axial compliance (Section \ref{ssec:detail_fem_joint_axial}).
The analysis of bending and axial stiffness effects enables the establishment of comprehensive specifications for the flexible joints.
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.
\section{Bending and Torsional Stiffness}
\label{sec:orgcea815a}
\label{sec:joints_rot_stiffness}
\label{sec:org582c93a}
\label{ssec:detail_fem_joint_bending}
Because of bending stiffness of the flexible joints, the forces applied by the struts are no longer aligned with the struts (additional forces applied by the ``spring force'' of the flexible joints).
The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction.
This additional spring forces can affect system dynamics.
In this section, we wish to study the effect of the rotation flexibility of the nano-hexapod joints.
\begin{itemize}
\item To simplify the analysis, the micro-station is considered rigid, and only the nano-hexapod is considered with:
\begin{itemize}
\item 1dof actuators, k=1N/um, without parallel stiffness to the force sensors
\end{itemize}
\item The bending stiffness of all joints are varied and the dynamics is identified
\end{itemize}
To isolate and quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified single-degree-of-freedom actuators (stiffness \(1\,N/\mu m\)) without parallel force sensor stiffness.
Joint bending stiffness was varied from 0 (ideal case) to 500 Nm/rad.
HAC plant (transfer function from f to dL, as measured by the external metrology):
\begin{itemize}
\item It increase the coupling at low frequency, but is kept to small values for realistic values of the bending stiffness (Figure \ref{fig:detail_fem_joints_bending_stiffness_hac_plant})
\item Bending stiffness does not impact significantly the HAC plant.
The added stiffness increases the frequency of the suspension modes
Condition in \cite{mcinroy02_model_desig_flexur_joint_stewar} to have forces aligned with the struts when considering rotational stiffness: kr << k*l\^{}2
For the current nano hexapod configuration, it correspond to << 9000 Nm/rad.
This may be an issue for soft nano-hexapod (for instance k = 1e4 => << 90) => have to design very soft flexible joints.
Here, having relatively stiff actuators render this condition easier to achieve.
\end{itemize}
Analysis of the plant dynamics reveals two significant effects.
For the transfer function from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\), bending stiffness increases low-frequency coupling, though this remains small for realistic stiffness values (Figure \ref{fig:detail_fem_joints_bending_stiffness_hac_plant}).
In \cite{mcinroy02_model_desig_flexur_joint_stewar}, it is established that forces remain effectively aligned with the struts when the flexible joint bending stiffness is much small than the actuator stiffness multiplied by the square of the strut length.
For the nano-hexapod, this corresponds to having the bending stiffness much lower than 9000 Nm/rad.
This condition is more readily satisfied with the relatively stiff actuators selected, and could be problematic for softer Stewart platforms.
IFF Plant:
\begin{itemize}
\item Having bending stiffness adds complex conjugate zero at low frequency (Figure \ref{fig:detail_fem_joints_bending_stiffness_iff_plant})
\item Similar to having a stiffness in parallel to the struts (i.e., to the force sensor).
This can be explained since even if the force sensor is removed (i.e. zero axial stiffness of the strut), the strut will still act as a spring between the mobile and fixed plates because of the bending stiffness of the flexible joints.
The frequency of the zero gives an idea of the stiffness contribution of the flexible joint bending stiffness
\item They therefore impose limitation for decentralized IFF, as discussed in \cite{preumont07_six_axis_singl_stage_activ}
\item This can be seen in the root locus plot of Figure \ref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}
\end{itemize}
For the force sensor plant, bending stiffness introduces complex conjugate zeros at low frequency (Figure \ref{fig:detail_fem_joints_bending_stiffness_iff_plant}).
This behavior resembles having parallel stiffness to the force sensor as was the case with the APA300ML (see Figure \ref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}).
However, this time the parallel stiffness does not comes from the considered strut, but from the bending stiffness of the flexible joints of the other five struts.
This characteristic impacts the achievable damping using decentralized Integral Force Feedback \cite{preumont07_six_axis_singl_stage_activ}.
This is confirmed by the Root Locus plot in Figure \ref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}.
This effect becomes less significant when using the selected APA300ML actuators (Figure \ref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}), which already incorporate parallel stiffness by design which is higher than the one induced by flexible joint stiffness.
A parallel analysis of torsional stiffness revealed similar dynamic effects, though these proved less critical for system performance.
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
@ -714,13 +662,6 @@ The frequency of the zero gives an idea of the stiffness contribution of the fle
\caption{\label{fig:detail_fem_joints_bending_stiffness_plants}Effect of bending stiffness of the flexible joints on the plant dynamics. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_bending_stiffness_hac_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_bending_stiffness_iff_plant})}
\end{figure}
However, as the APA300ML was chosen for the actuator, stiffness are already present in parallel to the force sensors:
\begin{itemize}
\item The dynamics is computed again for all considered values of the bending stiffnesses with the 2DoF model of the APA300ML
\item Root locus for decentralized IFF are shown in Figure \ref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}.
Now the effect of bending stiffness has little effect on the attainable damping, as its contribution as ``parallel stiffness'' is small compared to the parallel stiffness already present in the APA300ML.
\end{itemize}
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
@ -737,32 +678,29 @@ Now the effect of bending stiffness has little effect on the attainable damping,
\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}
Conclusion:
\begin{itemize}
\item Similar results for torsional stiffness, but less important
\item thanks to the use of the APA, the requirements in terms of bending stiffness are less stringent
\end{itemize}
\section{Axial Stiffness}
\label{sec:org0768c98}
\label{sec:joints_trans_stiffness}
\label{sec:org05206a1}
\label{ssec:detail_fem_joint_axial}
\begin{itemize}
\item Adding flexibility between the actuation point and the measurement point / point of interest is always detrimental for the control performances.
This is verified, and the goal is to estimate the minimum axial stiffness that the flexible joints should have
\item Here, the mass of the strut should be considered.
It is set to 112g as specified in the APA300ML specification sheet.
The limited axial stiffness (\(k_a\)) of flexible joints introduces an additional compliance between the actuation point and the measurement point.
As explained in \cite[, chapter 6]{preumont18_vibrat_contr_activ_struc_fourt_edition} and in \cite{rankers98_machin} (effect called ``actuator flexibility''), such intermediate flexibility invariably degrades control performance.
Therefore, determining the minimum acceptable axial stiffness that maintains nano-hexapod performance becomes crucial.
\item Transfer functions are estimated for several axial stiffnesses (Figure \ref{fig:detail_fem_joints_axial_stiffness_plants})
\item IFF plant is not much affected (Figure \ref{fig:detail_fem_joints_axial_stiffness_iff_plant}).
Confirmed by the root locus plot of Figure \ref{fig:detail_fem_joints_axial_stiffness_iff_locus}
\item ``HAC'' plant:
\begin{itemize}
\item Additional modes at high frequency corresponding to internal modes of the struts.
It adds coupling to the plant.
This is confirmed by computed the RGA-number for the damped plant (i.e. after applying decentralized IFF) in Figure \ref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}
\end{itemize}
\end{itemize}
The analysis incorporates the strut mass (112g per APA300ML) to accurately model internal resonance effects.
A parametric study was conducted by varying the axial stiffness from \(1\,N/\mu m\) (matching actuator stiffness) to \(1000\,N/\mu m\) (approximating rigid behavior).
The resulting frequency responses (Figure \ref{fig:detail_fem_joints_axial_stiffness_plants}) reveal distinct effects on system dynamics.
The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both frequency response data (Figure \ref{fig:detail_fem_joints_axial_stiffness_iff_plant}) and root locus analysis (Figure \ref{fig:detail_fem_joints_axial_stiffness_iff_locus}).
However, the externally measured (HAC) plant demonstrates significant effects: internal strut modes appear at high frequencies, introducing substantial cross-coupling between axes.
This coupling is quantified through RGA analysis of the damped system (Figure \ref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance.
Above this resonance frequency, two critical limitations emerge.
First, the system exhibits strong coupling between control channels, making decentralized control strategies ineffective.
Second, control authority diminishes significantly near the resonant frequencies.
These effects fundamentally limit achievable control bandwidth, making high axial stiffness essential for system performance.
Based on this analysis, an axial stiffness specification of \(100\,N/\mu m\) was established for the nano-hexapod joints.
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
@ -780,12 +718,6 @@ This is confirmed by computed the RGA-number for the damped plant (i.e. after ap
\caption{\label{fig:detail_fem_joints_axial_stiffness_plants}Effect of axial stiffness of the flexible joints on the plant dynamics. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_axial_stiffness_hac_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_axial_stiffness_iff_plant})}
\end{figure}
Integral force feedback
\begin{itemize}
\item[{$\square$}] Maybe show the damped plants instead?
\item[{$\square$}] Root Locus: not a lot of effect
\end{itemize}
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
@ -802,42 +734,13 @@ Integral force feedback
\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}
Conclusion:
\begin{itemize}
\item The axial stiffness of the flexible joints should be maximized to limit additional coupling at high frequency that may negatively impact the achievable bandwidth
\item It should be much higher than the stiffness of the actuator
\item For the nano-hexapod 100N/um is a reasonable axial stiffness specification
\item Above the resonance frequency linked to the limited axial stiffness of the flexible joint, the system becomes coupled and impossible to control
\item Also, loose control authority at the frequency of the zero
\end{itemize}
\section{Specifications and Design flexible joints}
\label{sec:org2310ef0}
\label{ssec:detail_fem_joint_specs}
\section{Obtained design / Specifications}
\label{sec:org17da38c}
\begin{itemize}
\item Summary of specifications (Table \ref{tab:detail_fem_joints_specs})
\item Explain choice of geometry:
\begin{itemize}
\item x and y rotations are coincident
\item stiffness can be easily tuned
\item high axial stiffness
\end{itemize}
\item Explain how it is optimized:
\begin{itemize}
\item Extract stiffnesses from FEM
\item Parameterized model in the FE software
\item Quick optimization: (few iterations, could probably increase more the axial stiffness)
\begin{itemize}
\item There is a trade off between high axial stiffness and low bending/torsion stiffness
\item Also check the yield strength
\end{itemize}
\end{itemize}
\item Show obtained geometry Figure \ref{fig:detail_fem_joints_design}:
\begin{itemize}
\item ``neck'' size: 0.25mm
\end{itemize}
\item Characteristics of the flexible joints obtained from FEA are summarized in Table \ref{tab:detail_fem_joints_specs}
\end{itemize}
The design of flexible joints for precision applications requires careful consideration of multiple mechanical characteristics.
Critical specifications include sufficient bending stroke to ensure long-term operation below yield stress, high axial stiffness for precise positioning, low bending and torsional stiffnesses to minimize parasitic forces, adequate load capacity, and well-defined rotational axes.
Based on the dynamic analysis presented in previous sections, quantitative specifications were established and are summarized in Table \ref{tab:detail_fem_joints_specs}.
\begin{table}[htbp]
\caption{\label{tab:detail_fem_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model}
@ -855,6 +758,15 @@ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
\end{tabularx}
\end{table}
Among various possible flexible joint architectures, the design shown in Figure \ref{fig:detail_fem_joints_design} was selected for three key advantages.
First, the geometry creates coincident x and y rotation axes, ensuring well-defined kinematic behavior through precise definition of the system's Jacobian matrix.
Second, the design allows easy tuning of different directional stiffnesses through a limited number of geometric parameters.
Third, the architecture inherently provides high axial stiffness while maintaining the required compliance in rotational degrees of freedom.
The joint geometry was optimized through parametric finite element analysis.
The optimization process revealed an inherent trade-off between maximizing axial stiffness and achieving sufficiently low bending/torsional stiffness, while maintaining material stresses within acceptable limits.
The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through finite element analysis and documented in Table \ref{tab:detail_fem_joints_specs}.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
@ -872,17 +784,12 @@ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
\end{figure}
\section{Validation with the Nano-Hexapod}
\label{sec:orgd711eb0}
\label{sec:org4e972fc}
\label{ssec:detail_fem_joint_validation}
To validate the designed flexible joint:
\begin{itemize}
\item FEM: modal reduction
two interface frames are defined (Figure \ref{fig:detail_fem_joints_frames})
\item additional 6 modes are extracted: size of reduced order mass and stiffness matrices: \(18 \times 18\)
\item Imported in the multi-body model
\item The transfer functions from forces and torques applied between frames \(\{F\}\) and \(\{M\}\) to the relative displacement/rotations of the two frames is extracted.
\item The stiffness characteristics of the flexible joint is estimated from the low frequency gain of the obtained transfer functions. Same values are obtained with the reduced order model and the FEM.
\end{itemize}
The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from finite element analysis.
This high-fidelity representation was created by defining two interface frames (Figure \ref{fig:detail_fem_joints_frames}) and extracting six additional modes, resulting in reduced-order mass and stiffness matrices of dimension \(18 \times 18\).
The computed transfer functions from actuator forces to both force sensor measurements (\(\bm{f}\) to \(\bm{f}_m\)) and external metrology (\(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\)) demonstrate dynamics consistent with predictions from earlier analyses (Figure \ref{fig:detail_fem_joints_fem_vs_perfect_plants}), thereby validating the joint design.
\begin{figure}[htbp]
\centering
@ -890,44 +797,13 @@ two interface frames are defined (Figure \ref{fig:detail_fem_joints_frames})
\caption{\label{fig:detail_fem_joints_frames}Defined frames for the reduced order flexible body. The two flat interfaces are considered rigid, and are linked to the two frames \(\{F\}\) and \(\{M\}\) both located at the center of the rotation.}
\end{figure}
Depending on which characteristic of the flexible joint is to be modelled, several DoFs can be taken into account:
\begin{itemize}
\item 2DoF (universal joint) \(k_f\)
\item 3DoF (spherical joint) taking into account torsion \(k_f\), \(k_t\)
\item 2DoF + axial stiffness \(k_f\), \(k_a\)
\item 3DoF + axial stiffness \(k_f\), \(k_t\), \(k_a\)
\item 6DoF (``bushing joint'') \(k_f\), \(k_t\), \(k_a\), \(k_s\)
\end{itemize}
While this detailed modeling approach provides high accuracy, it results in a significant increase in system model order.
The complete nano-hexapod model incorporates 240 states: 12 for the payload (6 DOF), 12 for the 2DOF struts, and 216 for the flexible joints (18 states for each of the 12 joints).
To improve computational efficiency, a low order representation was developed using simplified joint elements with selective compliance DoF.
Adding more degrees of freedom:
\begin{itemize}
\item can represent important features
\item adds model states that may not be relevant for the dynamics, and may complexity the simulations without adding much information
\end{itemize}
After testing different configurations, a good compromise was found for the modelling of the nano-hexapod flexible joints:
\begin{itemize}
\item bottom joints: \(k_f\) and \(k_a\)
\item top joints: \(k_f\), \(k_t\) and \(k_a\)
\end{itemize}
Talk about model order:
\begin{itemize}
\item with flexible joints: 252 states:
\begin{itemize}
\item 12 for the payload (6 dof)
\item 12 for the 2DoF struts
\item 216 DoF for the flexible joints (18*6*2)
\item 12 states for?
\end{itemize}
\item with 3dof and 4dof: 48 states
\begin{itemize}
\item 12 for the payload (6 dof)
\item 12 for the 2DoF struts
\item 12 states for the bottom joints
\item 12 states for the top joints
\end{itemize}
\end{itemize}
After evaluating various configurations, a compromise was achieved by modeling bottom joints with bending and axial stiffness (\(k_f\) and \(k_a\)), and top joints with bending, torsional, and axial stiffness (\(k_f\), \(k_t\) and \(k_a\)).
This simplification reduces the total model order to 48 states: 12 for the payload, 12 for the struts, and 24 for the joints (12 each for bottom and top joints).
While additional degrees of freedom could potentially capture more dynamic features, the selected configuration preserves essential behavioral characteristics while minimizing computational complexity.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@ -946,8 +822,20 @@ Talk about model order:
\end{figure}
\chapter*{Conclusion}
\label{sec:orgd540d5a}
\label{sec:org5382fe7}
\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 phase of the nano-hexapod.
The approach was first validated using an amplified piezoelectric actuator, where predicted dynamics showed excellent agreement with experimental measurements for both open and closed-loop behavior.
This validation established confidence in the method's ability to accurately predict component behavior within the broader system context.
The methodology was then successfully applied to optimize two critical components.
For the actuators, it enabled validation of the APA300ML selection while providing both high-fidelity and computationally efficient models for system simulation.
Similarly, for the flexible joints, the analysis of bending and axial stiffness effects led to clear specifications and an optimized design that balances competing mechanical requirements.
In both cases, the ability to seamlessly integrate finite element models into the multi-body framework proved essential for understanding component interactions and their impact on system-level dynamics.
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.
\printbibliography[heading=bibintoc,title={Bibliography}]
\end{document}