% Created 2025-02-26 Wed 15:42
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}

\input{preamble.tex}
\input{preamble_extra.tex}
\bibliography{nass-fem.bib}
\author{Dehaeze Thomas}
\date{\today}
\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}

\begin{document}

\maketitle
\tableofcontents

\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}

\chapter{Reduced order flexible bodies}
\label{sec:org5704c94}
\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.
However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be successfully integrated into multi-body models \cite{hatch00_vibrat_matlab_ansys}.
This combined multibody-FEA modeling approach presents significant advantages, as it enables the selective application of FEA modeling to specific elements while maintaining the computational efficiency of multi-body analysis for the broader system \cite{rankers98_machin}.

The investigation of this hybrid modeling approach is structured in three sections.
First, the fundamental principles and methodological approaches of this modeling framework are introduced (Section \ref{ssec:detail_fem_super_element_theory}).
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:orga74dca6}
\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.
These matrices are generated via modal reduction techniques, specifically through the application of component mode synthesis (CMS), thus establishing this design approach as a combined multibody-FEA methodology.

Standard FEA implementations typically involve thousands or even hundreds of thousands of DoF, rendering direct integration into multi-body simulations computationally prohibitive.
The objective of modal reduction is therefore to substantially decrease the number of DoF while preserving the essential dynamic characteristics of the component.

The procedure for implementing this reduction involves several distinct stages.
Initially, the component is modeled in a finite element software with appropriate material properties and boundary conditions.
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.

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 transforms the extensive FEA degrees of freedom into a significantly reduced set of retained degrees of freedom.
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 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.

\begin{equation}\label{eq:detail_fem_model_order}
m = 6 \times n + p
\end{equation}

\section{Example with an Amplified Piezoelectric Actuator}
\label{sec:org3e7c2ec}
\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}.
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 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.

\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_fem_apa95ml_picture.png}
\captionof{figure}{\label{fig:detail_fem_apa95ml_picture}Picture of the APA95ML}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\linewidth}
\centering
\begin{tabularx}{0.7\linewidth}{Xc}
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
Nominal Stroke & \(100\,\mu m\)\\
Blocked force & \(2100\,N\)\\
Stiffness & \(21\,N/\mu m\)\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage}
\paragraph{Finite Element Model}
\label{sec:org491eeae}

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.

\begin{table}[htbp]
\caption{\label{tab:detail_fem_material_properties}Material properties used for FEA modal reduction model. \(E\) is the Young's modulus, \(\nu\) the Poisson ratio and \(\rho\) the material density}
\centering
\begin{tabularx}{0.7\linewidth}{lXXX}
\toprule
 & \(E\) & \(\nu\) & \(\rho\)\\
\midrule
Stainless Steel & \(190\,GPa\) & \(0.31\) & \(7800\,\text{kg}/m^3\)\\
Piezoelectric Ceramics (PZT) & \(49.5\,GPa\) & \(0.31\) & \(7800\,\text{kg}/m^3\)\\
\bottomrule
\end{tabularx}
\end{table}

The definition of interface frames, or ``remote points'', constitute 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.

Six additional modes were considered, resulting in total model order of \(48\).
The modal reduction procedure was then executed, yielding the reduced mass and stiffness matrices that form the foundation of the component's representation in the multi-body simulation environment.

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_fem_apa95ml_mesh.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_mesh}Obtained mesh and "remote points"}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_fem_apa_modal_schematic.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa_model_schematic}Inclusion in multi-body model}
\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_modal_schematic}).}
\end{figure}

\paragraph{Super Element in the Multi-Body Model}
\label{sec:org29dd028}

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.
Frame \(\{4\}\) was connected to the ``world'' frame, while frame \(\{6\}\) was coupled to a vertically guided payload.
In this example, two piezoelectric stacks were used for actuation while one piezoelectric stack was used as a force sensor.
Therefore, a force source \(F_a\) operating between frames \(\{3\}\) and \(\{2\}\) was used, while a displacement sensor \(d_L\) between frames \(\{1\}\) and \(\{7\}\) was used for the sensor stack.
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:org1329f1a}

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}.

From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation between relative displacement \(d_L\) of the sensor stack and generated voltage \(V_s\) is given by \eqref{eq:detail_fem_dl_to_vs}.

\begin{equation}\label{eq:detail_fem_dl_to_vs}
  V_s = g_s \cdot d_L, \quad g_s = \frac{d_{33}}{\epsilon^T s^D n}
\end{equation}

From \cite{fleming10_integ_strain_force_feedb_high} the relation between the force \(F_a\) and the applied voltage \(V_a\) is given by \eqref{eq:detail_fem_va_to_fa}.

\begin{equation}\label{eq:detail_fem_va_to_fa}
  F_a = g_a \cdot V_a, \quad g_a = d_{33} n k_a, \quad k_a = \frac{c^{E} A}{L}
\end{equation}

Unfortunately, it is difficult to know exactly which material is used in the amplified piezoelectric actuator\footnote{The manufacturer of the APA95ML was not willing to share the piezoelectric material properties of the stack.}.
However, based on the available properties of the stacks in the data-sheet (summarized in Table \ref{tab:detail_fem_stack_parameters}), the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.

\begin{table}[htbp]
\caption{\label{tab:detail_fem_stack_parameters}Stack Parameters}
\centering
\begin{tabularx}{0.4\linewidth}{Xcc}
\toprule
Parameter & Unit & Value\\
\midrule
Nominal Stroke & \(\mu m\) & 20\\
Blocked force & \(N\) & 4700\\
Stiffness & \(N/\mu m\) & 235\\
Voltage Range & \(V\) & -20 to 150\\
Capacitance & \(\mu F\) & 4.4\\
Length & \(mm\) & 20\\
Stack Area & \(mm^2\) & 10x10\\
\bottomrule
\end{tabularx}
\end{table}

The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table \ref{tab:test_apa_piezo_properties}.
From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained.

\begin{table}[htbp]
\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators sensitivities}
\centering
\begin{tabularx}{1\linewidth}{ccX}
\toprule
\textbf{Parameter} & \textbf{Value} & \textbf{Description}\\
\midrule
\(d_{33}\) & \(680 \cdot 10^{-12}\,m/V\) & Piezoelectric constant\\
\(\epsilon^{T}\) & \(4.0 \cdot 10^{-8}\,F/m\) & Permittivity under constant stress\\
\(s^{D}\) & \(21 \cdot 10^{-12}\,m^2/N\) & Elastic compliance understand constant electric displacement\\
\(c^{E}\) & \(48 \cdot 10^{9}\,N/m^2\) & Young's modulus of elasticity\\
\(L\) & \(20\,mm\) per stack & Length of the stack\\
\(A\) & \(10^{-4}\,m^2\) & Area of the piezoelectric stack\\
\(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\
\bottomrule
\end{tabularx}
\end{table}

\paragraph{Identification of the APA Characteristics}
\label{sec:org5512e6c}

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.

The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure \ref{fig:detail_fem_apa95ml_compliance}), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames.
The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML.
A value of \(23\,N/\mu m\) was found which is close to the specified stiffness in the datasheet of \(k = 21\,N/\mu m\).

The multi-body model predicted a resonant frequency under block-free conditions of \(2024\,\text{Hz}\) (Figure \ref{fig:detail_fem_apa95ml_compliance}), which is in agreement with the nominal specification of \(2000\,\text{Hz}\).

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_fem_apa95ml_compliance.png}
\caption{\label{fig:detail_fem_apa95ml_compliance}Estimated compliance of the APA95ML}
\end{figure}

In order to estimate the stroke of the APA95ML, first the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, needs to be determined.
This characteristic was quantified through analysis of the transfer function relating horizontal stack motion to vertical actuator displacement, from which an amplification factor of \(1.5\) was derived.

The piezoelectric stacks, exhibiting a typical strain response of \(0.1\,\%\) relative to their length (here equal to \(20\,mm\)), produce an individual nominal stroke of \(20\,\mu m\) (see data-sheet of the piezoelectric stacks on Table \ref{tab:detail_fem_stack_parameters}, page \pageref{tab:detail_fem_stack_parameters}).
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 \(120\,\mu m\).

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:org8627abc}
\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.

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 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 amplification stage providing a gain factor 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).

\begin{figure}[htbp]
\begin{subfigure}{0.34\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_bench_picture.png}
\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}
\paragraph{Comparison of the dynamics}
\label{sec:orgb3fa207}

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 most used ones are impulses (particularly suited to modal analysis), steps, random noise signals, and multi-sine excitations.
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.
From the obtained input and output data, the frequency response functions were derived.
To improve the quality of the obtained frequency domain data, averaging and windowing were used \cite[, chap. 13]{pintelon12_system_ident}..

The obtained frequency response functions from \(V_a\) to \(V_s\) and to \(y\) are compared with the theoretical predictions derived from the multi-body model in Figure \ref{fig:detail_fem_apa95ml_comp_plant}.

The difference in phase between the model and the measurements can be attributed to the sampling time of \(0.1\,ms\) and to additional delays induced by electronic instrumentation related to the interferometer.
The presence of a non-minimum phase zero in the measured system response (Figure \ref{fig:detail_fem_apa95ml_comp_plant_sensor}), shall be addressed during the experimental phase.

Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could be further refined through calibration against the experimental data.

\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_comp_plant_actuator.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $y$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_comp_plant_sensor.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_sensor}from $V_a$ to $V_s$}
\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})}
\end{figure}

\paragraph{Integral Force Feedback with APA}
\label{sec:org182828d}

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.

The IFF controller implementation, defined in equation \ref{eq:detail_fem_iff_controller}, incorporated a tunable gain parameter \(g\) and was designed to provide integral action near the system resonances and to limit the low frequency gain using an high pass filter.

\begin{equation}\label{eq:detail_fem_iff_controller}
  K_{\text{IFF}}(s) = \frac{g}{s + 2\cdot 2\pi} \cdot \frac{s}{s + 0.5 \cdot 2\pi}
\end{equation}

The theoretical damped dynamics of the closed-loop system was analyzed through using the model by computed the root locus plot shown in Figure \ref{fig:detail_fem_apa95ml_iff_root_locus}.
For experimental validation, six gain values were tested: \(g = [0,\,10,\,50,\,100,\,500,\,1000]\).
The measured frequency responses for each gain configuration were compared with model predictions, as presented in Figure \ref{fig:detail_fem_apa95ml_damped_plants}.

The close agreement between experimental measurements and theoretical predictions across all gain configurations demonstrates the model's capability to accurately predict both open-loop and closed-loop system dynamics, thereby validating its utility for control system design and analysis.

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root Locus plot}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_damped_plants.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_damped_plants}Damped plants}
\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.}
\end{figure}

\section*{Conclusion}
\label{sec:org7af3b1c}
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.

While this modeling approach provides accurate predictions of component behavior, the resulting model order can become prohibitively high for practical time-domain simulations.
This is exemplified by the nano-hexapod configuration, where the implementation of six Amplified Piezoelectric Actuators, each modeled with 48 degrees of freedom, yields 288 degrees of freedom only for the actuators.
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}
\label{sec:orgcb23435}
\label{sec:detail_fem_actuator}
Goals:
\begin{itemize}
\item Based on dynamical models and previous studies, extract specifications for the actuators to be included in the nano-hexapod.
Then choose the most appropriate actuator based on specifications (Section \ref{ssec:detail_fem_actuator_specifications})
\item Model this actuator accurately using a ``reduced order flexible body'' to check the dynamics and validate the choice of actuator
and validate this choice with simulations
\item Development of a 2DoF model for lower order models (i.e. for simulations)
\end{itemize}
\section{Choice of the Actuator based on Specifications}
\label{sec:org6a6861c}
\label{ssec:detail_fem_actuator_specifications}

From previous analysis:
\begin{itemize}
\item Actuator stiffness has major impact on the system dynamics and performances due to several factors:
\begin{itemize}
\item Spindle rotation: modification of plant dynamics and coupling increase due to Gyroscopic effects
This require to have stiffness above \textasciitilde{}
\item Limited micro-station compliance / complex dynamics:
The actuator stiffness should be small enough such that the suspension modes of the nano-hexapod are below the problematic modes of the micro-stations.
\item There is therefore an intermediate stiffness that is foreseen to give the best compromise, and it is around \(1\,N/\mu m\)
\end{itemize}
\item HAC-LAC strategy:
Actuator must include a force sensor
Because of the rotation, some stiffness should be present in parallel to the force sensor
\item Limited space:
As the maximum height of the nano-hexapod is 95mm, and each strut has a flexible joint at each end, it is estimated that the maximum height of the actuator should be less than 50mm
\item Stroke:
The stroke of the each actuator should be large enough such that the nano-hexapod mobility exceed the micro-station positioning errors.
Some margins should be included for mounting errors, and further flexibility of the system (for instance to perform scans with the nano-hexapod, or to align the point of interest with the rotation axis)
\end{itemize}

Actuator specifications:
\begin{itemize}
\item Height (<50mm)
\item Stroke (\textasciitilde{}100um)
\item Stiffness (0.1-1 N/um)
\item Blocked force?
\item Force sensor
\end{itemize}

Options:
\begin{itemize}
\item Two main options: piezoelectric actuators and Lorentz actuator (also known as Voice coil actuators).
Variable reluctance actuators were not considered, even though they have better efficiency than voice coil actuators, they are non linear and induce additional control complexity.
\item Voice coil + relatively soft flexible guiding (1N/um):
\begin{itemize}
\item required force \textasciitilde{}100N for 100um correction
This constant force/current would induce large thermal loads, that may negatively impact system's stability
Advantages of voice coil (longer strokes than piezo + allow for very low stiffness in the direction of actuation, extremely linear for high performance feedforward) are not used here.
\end{itemize}
\item Piezoelectric stack actuators:
\begin{itemize}
\item PZT: stroke \textasciitilde{}0.1\% of its length.
\item 50mm length => 50um stroke which is barely enough
\item Extremely stiff, in the order of \(100\,N/\mu m\), which is not wanted here.
\end{itemize}
\item Amplified Piezoelectric Actuator:
\begin{itemize}
\item shell is used to pre-stress the piezoelectric stacks and amplify the motion (roughly by the ratio of the width over the height)
\item This also reduce the stiffness in the direction of motion
\item This make this design quick compact in the direction of motion (i.e. in height)
\item When several stacks are used, one of them can be used as a force sensor, which is therefore very well collocated with the actuators
\item Therefore, this actuator is well suited for decentralized IFF, already applied for a Stewart platform with APA \cite{hanieh03_activ_stewar}
\end{itemize}
\end{itemize}


\begin{figure}[htbp]
\begin{subfigure}{0.25\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/detail_fem_voice_coil_picture.jpg}
\end{center}
\subcaption{\label{fig:detail_fem_voice_coil_picture}Voice Coil}
\end{subfigure}
\begin{subfigure}{0.25\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/detail_fem_piezo_picture.jpg}
\end{center}
\subcaption{\label{fig:detail_fem_piezo_picture}Piezoelectric stack}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{center}
\includegraphics[scale=1,height=3.5cm]{figs/detail_fem_fpa_picture.jpg}
\end{center}
\subcaption{\label{fig:detail_fem_fpa_picture}Amplified Piezoelectric Actuator}
\end{subfigure}
\caption{\label{fig:detail_fem_actuator_pictures}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}).}
\end{figure}

Based on previous analysis, it was decided to use amplified piezoelectric actuators for the nano-hexapod.
Table \ref{tab:detail_fem_piezo_act_models}: compares few models that fulfill specifications.
It was decided to go for the APA300ML (shown in Figure \ref{fig:detail_fem_apa300ml_picture}).
One reason is that we already had experience with APA from Cedrat technologies, and the Finite Element Model was validated experimentally, so we are confident to model the APA300ML with FEA and include it in the NASS model for validation.


\begin{itemize}
\item Talk about piezoelectric actuator? bandwidth? noise?
\item Resolution: really depends on the electrical noise (induced by DAC and voltage amplifier).
They will be chosen appropriately
\end{itemize}

\begin{table}[htbp]
\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the nano-hexapod}
\centering
\scriptsize
\begin{tabularx}{0.9\linewidth}{Xccccc}
\toprule
\textbf{Specification} & APA150M & \textbf{APA300ML} & APA400MML & FPA-0500E-P & FPA-0300E-S\\
\midrule
Stroke  \(> 100\, [\mu m]\) & 187 & 304 & 368 & 432 & 240\\
Stiffness  \(\approx 1\, [N/\mu m]\) & 0.7 & 1.8 & 0.55 & 0.87 & 0.58\\
Resolution  \(< 2\, [nm]\) & 2 & 3 & 4 &  & \\
Blocked Force  \(> 100\, [N]\) & 127 & 546 & 201 & 376 & 139\\
Height  \(< 50\, [mm]\) & 22 & 30 & 24 & 27 & 16\\
\bottomrule
\end{tabularx}
\end{table}

\section{APA300ML - Reduced Order Flexible Body}
\label{sec:org56a3ff1}
\label{ssec:detail_fem_actuator_apa300ml}

To validate the choice of the APA300ML (Shown in Figure \ref{fig:detail_fem_apa300ml_picture}):
\begin{itemize}
\item the APA300ML is modeled using a Finite Element Software
\item a \emph{super element} is exported and imported in Simscape where its dynamic is studied
\item similarly to what was done with the APA95ML, frames defined for the \emph{super element} are shown in figure \ref{fig:detail_fem_apa300ml_frames}
\end{itemize}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa300ml_picture.jpg}
\end{center}
\subcaption{\label{fig:detail_fem_apa300ml_picture}Picture of the APA300ML}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa300ml_frames.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa300ml_frames}FEM of the APA300ML}
\end{subfigure}
\caption{\label{fig:detail_fem_apa300ml}Amplified Piezoelectric Actuator APA300ML. Picture shown in (\subref{fig:detail_fem_apa300ml_picture}). Frames (or ``remote points'') used for the modal reduction are shown in (\subref{fig:detail_fem_apa300ml_frames}).}
\end{figure}

\begin{itemize}
\item For this reduced order model, 7 frames are defined and 120 additional modes are modelled for a total matrix size of 162.
\item This is very large and will not be practical for simulations, but the best model accuracy was wanted for validation
\item The blue frames are used to model the force sensor stack: the relative motion between the two frame is measured
\item The red frames are used to model the two actuator stacks: \emph{internal force} are added
\item One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G).
\item The link between mechanical properties and electrical properties was discussed in Section \ref{ssec:detail_fem_super_element_validation}.
As the stacks are the same between the APA300ML and the APA95ML, the values estimated for \(g_a\) and \(g_s\) are used for the APA300ML.
\end{itemize}

\section{Simpler 2DoF Model of the APA300ML}
\label{sec:orgfabc6b8}
\label{sec:apa_model}
\begin{itemize}
\item \emph{super-element} order is quite large, and therefore not practical for simulations
\item the goal here is to develop a low order model, that still represents wanted characteristics of the APA300ML:
\begin{itemize}
\item axial stiffness
\item actuator and force sensor characteristics
\end{itemize}
\item what is not modelled:
\begin{itemize}
\item higher order modes
\item the flexibility of the APA in the other directions
\end{itemize}
\item Therefore this model can be useful for simulations as it contains a very limited number of states, but when more complex dynamics of the APA is to be modelled, a flexible model will be used.
\end{itemize}
\paragraph{2DoF Model}
\label{sec:org5962dd3}

The model is adapted from \cite{souleille18_concep_activ_mount_space_applic}.

It can be decomposed into three components:
\begin{itemize}
\item the shell whose axial properties are represented by \(k_1\) and \(c_1\)
\item the actuator stacks whose contribution to the axial stiffness is represented by \(k_a\) and \(c_a\).
The force source \(f\) represents the axial force induced by the force sensor stacks.
The sensitivity \(g_a\) (in \(N/m\)) is used to convert the applied voltage \(V_a\) to the axial force \(f\)
\item the sensor stack whose contribution to the axial stiffness is represented by \(k_e\) and \(c_e\).
A sensor measures the stack strain \(d_e\) which is then converted to a voltage \(V_s\) using a sensitivity \(g_s\) (in \(V/m\))
\end{itemize}

Such a simple model has some limitations:
\begin{itemize}
\item it only represents the axial characteristics of the APA as it is modeled as infinitely rigid in the other directions
\item some physical insights are lost, such as the amplification factor and the real stress and strain in the piezoelectric stacks
\item the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear
\end{itemize}

The main advantage is that this model is very simple, only adds 4 states

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_fem_apa_2dof_model.png}
\caption{\label{fig:detail_fem_apa_2dof_model}Schematic of the 2DoF model of the Amplified Piezoelectric Actuator}
\end{figure}

\paragraph{Parameter Tuning}
\label{sec:org7bd1971}
9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:detail_fem_apa_2dof_model}) well represents the identified dynamics using the FEM.

\begin{itemize}
\item Mass is 5kg (similar to the test bench)
\item Tune the parameters:
\begin{itemize}
\item From the first zero of the transfer function from Va to Vs, k1 and c1 are tuned
\item From the first pole of the transfer function from Va to y, ka, ca, ke, ce are tuned
\item because the actuator and sensor stacks are physically the same, we suppose
Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics.
Therefore, we have \(k_e = 2 k_a\) and \(c_e = 2 c_a\) as the actuator stack is composed of two stacks in series.
\item In the last step, \(g_s\) and \(g_a\) for the 2DoF motion can be tuned to match the gain of the transfer functions extracted from the FEM
\item Found parameters are summarized in Table \ref{tab:detail_fem_apa300ml_2dof_parameters}
\end{itemize}
\item Comparison of the transfer functions extracted from the high order flexible model with the 4th order (2DoF) model is done in Figure \ref{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}.
Good match is obtained.
Of course, higher order modes are not represented by the 2DoF model, nor the limited stiffness in the other directions.
\end{itemize}

\begin{table}[htbp]
\caption{\label{tab:detail_fem_apa300ml_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
\centering
\begin{tabularx}{0.3\linewidth}{cc}
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
\(k_1\) & \(0.30\,N/\mu m\)\\
\(k_e\) & \(4.3\, N/\mu m\)\\
\(k_a\) & \(2.15\,N/\mu m\)\\
\(c_1\) & \(18\,Ns/m\)\\
\(c_e\) & \(0.7\,Ns/m\)\\
\(c_a\) & \(0.35\,Ns/m\)\\
\(g_a\) & \(2.7\,N/V\)\\
\(g_s\) & \(0.53\,V/\mu m\)\\
\bottomrule
\end{tabularx}
\end{table}

\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa300ml_comp_fem_2dof_actuator.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}from $V_a$ to $d_i$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa300ml_comp_fem_2dof_force_sensor.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}from $V_a$ to $V_s$}
\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})}
\end{figure}

\section{Electrical characteristics of the APA}
\label{sec:org5cdd335}

\begin{itemize}
\item Mechanical equations and electrical equations are coupled
\item This means for instance, that the stiffness of the piezoelectric stack (i.e. the APA) depends on the electrical boundaries of the stacks:
\begin{itemize}
\item Short circuited stacks are less stiff than open-circuited ones
\item This effect is quite small: example with the APA95ML (Figure \ref{fig:detail_fem_apa95ml_effect_electrical_boundaries})
transfer function from Va to di are estimated with the force sensor stack being short circuited or open-circuited.
\end{itemize}
\item In the model used, the electrical phenomena are not modelled.
But as this effect is small, it should be fine
\item The electrical characteristics of the APA are very important both from the voltage amplifier side and the ADC measuring the force sensor voltage.
This will be discussed in chapter ``instrumentation''
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_fem_apa95ml_effect_electrical_boundaries.png}
\caption{\label{fig:detail_fem_apa95ml_effect_electrical_boundaries}Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency}
\end{figure}

\section{Validation with the Nano-Hexapod}
\label{sec:orgf89bb46}
NASS model + FEM model (or just 2DoF) of APA300ML => validation (based on what?)

\begin{itemize}
\item Compare 2DoF model and FEM (Figure \ref{fig:detail_fem_actuator_fem_vs_perfect_plants})
\begin{itemize}
\item HAC plant
\item IFF Plant
\item Very similar => can use 2nd order actuator models
\end{itemize}
\item Talk about model order
\begin{itemize}
\item 2DoF actuators: 24 states
\item FEM actuators:
here matrices have a size of 36
36*6+12 => \textasciitilde{}300
\end{itemize}
\end{itemize}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_actuator_fem_vs_perfect_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_actuator_fem_vs_perfect_iff_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\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}).}
\end{figure}



\chapter{Flexible Joint}
\label{sec:org8601117}
\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:orgd5923ee}

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}

\begin{figure}[htbp]
\begin{subfigure}{0.35\textwidth}
\begin{center}
\includegraphics[scale=1,height=5cm]{figs/detail_fem_joints_yang.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_yang}}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\begin{center}
\includegraphics[scale=1,height=5cm]{figs/detail_fem_joints_preumont.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_preumont}}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\begin{center}
\includegraphics[scale=1,height=5cm]{figs/detail_fem_joints_wire.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_wire}}
\end{subfigure}
\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}

\section{Bending and Torsional Stiffness}
\label{sec:orgf11a334}
\label{sec:joints_rot_stiffness}

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).

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}

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}

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}

\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_bending_stiffness_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_bending_stiffness_iff_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
\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}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_bending_stiffness_iff_locus_1dof.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}1DoF actuators}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_bending_stiffness_iff_locus_apa300ml.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}APA300ML actuators}
\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})}
\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:orgd08fa7c}
\label{sec:joints_trans_stiffness}

\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.

\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}

\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_axial_stiffness_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_axial_stiffness_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_axial_stiffness_iff_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
\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}
\includegraphics[scale=1,scale=0.9]{figs/detail_fem_joints_axial_stiffness_iff_locus.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_locus}Root Locus}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.9]{figs/detail_fem_joints_axial_stiffness_rga_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}RGA number}
\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})}
\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{Obtained design / Specifications}
\label{sec:org93383a9}

\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}

\begin{table}[htbp]
\caption{\label{tab:detail_fem_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model}
\centering
\begin{tabularx}{0.5\linewidth}{Xcc}
\toprule
 & \textbf{Specification} & \textbf{FEM}\\
\midrule
Axial Stiffness \(k_a\) & \(> 100\,N/\mu m\) & 94\\
Shear Stiffness \(k_s\) & \(> 1\,N/\mu m\) & 13\\
Bending Stiffness \(k_f\) & \(< 100\,Nm/\text{rad}\) & 5\\
Torsion Stiffness \(k_t\) & \(< 500\,Nm/\text{rad}\) & 260\\
Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
\bottomrule
\end{tabularx}
\end{table}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_fem_joints_3d_view.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_3d_view}3D view}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_fem_joint_dimensions.png}
\end{center}
\subcaption{\label{fig:detail_fem_joint_dimensions}Key dimensions}
\end{subfigure}
\caption{\label{fig:detail_fem_joints_design}Designed flexible joints.}
\end{figure}

\section{Validation with the Nano-Hexapod}
\label{sec:org9751c8e}

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}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_fem_joints_frames.png}
\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}

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}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_fem_vs_perfect_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_fem_vs_perfect_iff_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\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}).}
\end{figure}

\chapter*{Conclusion}
\label{sec:org57f9ca5}
\label{sec:detail_fem_conclusion}

\printbibliography[heading=bibintoc,title={Bibliography}]
\end{document}
