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\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},
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\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:detail_fem_super_element}
Goal:
\begin{itemize}
\item include parts from which dynamical properties are estimated from a FEM
\end{itemize}

Outline:
\begin{itemize}
\item Quick explanation of the theory
\item Explain the implementation with FEA software (Ansys) and Simscape
\item Experimental validation with an amplified piezoelectric actuator
\end{itemize}

\cite{rankers98_machin}
\cite{hatch00_vibrat_matlab_ansys}
\section{FEA Modal Reduction}
\label{ssec:detail_fem_super_element_theory}

\begin{itemize}
\item sub-components in the multi-body model as reduced order flexible bodies representing the component's modal behaviour with reduced mass and stiffness matrices obtained from finite element analysis (FEA) models
\item matrices were created from FEA models via modal reduction techniques, more specifically the component mode synthesis (CMS).
\item this makes this design approach a combined multibody-FEA technique.
\end{itemize}


\begin{itemize}
\item FEM: high number of DoF
\item goal: reduce number of DoF, allow to integrate in multi-body simulation
\end{itemize}

Procedure:
\begin{itemize}
\item model the part in FE software as usually by defining material properties, etc.
\item define frames for which we want to the multi-body model will then be able to interface with, and can be used to:
\begin{itemize}
\item connect other parts
\item apply forces and torques
\item measure motion between frames
\end{itemize}
\item perform the modal reduction technique from FEA (also called component mode synthesis or ``Craig-Bampton'' method) for the reduction of the high number of FEA degrees of freedom (DoF) to a smaller number of retained degrees of freedom
typically from hundred thousands to less than 100 DoF
\item the number of DoF is 6 times the number of defined frame + any number of additional DoF that we want to model
\(m = 6 \times n + p\)
\(n\) the number of frames, \(p\) the number of additional modes
\item then, it outputs \(m \times m\) reduced mass and stiffness matrices
\item in the multi-body model, the two reduced matrices can be used to model the part
\end{itemize}

\section{Validation of the Method}
\label{ssec:detail_fem_super_element_validation}
Validation with Amplified Piezoelectric Actuator, because:
\begin{itemize}
\item is a good candidate for the nano-hexapod (as will be explained in Section \ref{sec:detail_fem_apa})
\item had one in the lab for experimental testing (APA95ML, Figure \ref{fig:detail_fem_apa95ml_picture})
It is composed of several piezoelectric stacks (arranged horizontally, in blue), and a shell (in red) that amplifies the motion. The working direction of the APA95ML is vertical.
\item permits to model a mechanical structure (similar to a flexible joint), piezoelectric actuator and piezoelectric sensor
\end{itemize}

Quick explanation of APA:
\begin{itemize}
\item \cite{claeyssen07_amplif_piezoel_actuat}
\end{itemize}

\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}{0.8\linewidth}{Xcc}
\toprule
Parameter & Unit & Value\\
\midrule
Nominal Stroke & \(\mu m\) & 100\\
Blocked force & \(N\) & 1600\\
Stiffness & \(N/\mu m\) & 16\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage}
\paragraph{Finite Element Model}

\begin{itemize}
\item explain how the FEM is done:
\begin{itemize}
\item material properties (Table \ref{tab:detail_fem_material_properties})
\item mesh (Figure \ref{fig:detail_fem_apa95ml_fem})
\end{itemize}
\item explain piezoelectric materials:
\begin{itemize}
\item sensors
\item actuators
\end{itemize}
\item choice of frames (Figure \ref{fig:detail_fem_apa95ml_frames})
\begin{itemize}
\item 2 for each piezoelectric stack to measure strain and apply forces
\item 1 at the top, 1 at the bottom to connect to other elements
\end{itemize}
\item choose number of DoF => size of model
7 frames + 6 modes => order 48
\item perform the reduction: show the output reduced matrices
\end{itemize}

\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}

\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_mesh.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_mesh}Obtained mesh}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.85\linewidth]{figs/detail_fem_apa95ml_frames.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_frames}Defined frames}
\end{subfigure}
\caption{\label{fig:detail_fem_apa95ml_model}Finite element model of the APA95ML. Obtained mesh is shown in (\subref{fig:detail_fem_apa95ml_mesh}). Frames (or ``remote points'') used for the modal reduction are shown in (\subref{fig:detail_fem_apa95ml_frames}).}
\end{figure}

\paragraph{Super Element in the Multi-Body Model}

Model:
\begin{itemize}
\item Connect frame \(\{4\}\) to world frame and frame \(\{6\}\) to a 5.5kg mass, vertically guided
\item 2 actuator stacks, 1 sensor stack:
\begin{itemize}
\item force source between frames \(\{3\}\) and \(\{2\}\)
\item measured strain for force sensor by measuring the displacement between \(\{1\}\) and \(\{7\}\)
\end{itemize}
\item Input: internal force applied
\item Output: strain in the sensor stack
\item Issue: how to convert voltage to force and strain to voltage?
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.5\linewidth]{figs/detail_fem_apa_modal_schematic.png}
\caption{\label{fig:detail_fem_apa_model_schematic}Amplified Piezoelectric Actuator Schematic}
\end{figure}

Need to link the electrical domain (voltages, charges) with the mechanical domain (forces, strain).
To do so, ``actuator constant'' \(g_a\) and ``sensor constant'' \(g_s\) are used as shown in Figure \ref{fig:detail_fem_apa_model_schematic}.

A voltage \(V_a\) applied to the actuator stacks will induce an actuator force \(F_a\):
\begin{equation}
  \boxed{F_a = g_a \cdot V_a}
\end{equation}

A change of length \(dl\) of the sensor stack will induce a voltage \(V_s\):
\begin{equation}
  \boxed{V_s = g_s \cdot dl}
\end{equation}

In order to correctly model the piezoelectric actuator with Simscape, the values for \(g_a\) and \(g_s\) needs to be determined.
\begin{itemize}
\item \(g_a\): the ratio of the generated force \(F_a\) to the supply voltage \(V_a\) across the piezoelectric stack
\item \(g_s\): the ratio of the generated voltage \(V_s\) across the piezoelectric stack when subject to a strain \(\Delta h\)
\end{itemize}

\paragraph{Sensor and Actuator ``constants''}

The gains \(g_a\) and \(g_s\) were estimated from the physical properties of the piezoelectric stack material (summarized in Table \ref{tab:detail_fem_stack_parameters}).

\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}

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:test_apa_piezo_strain_to_voltage} and 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:test_apa_piezo_voltage_to_force}.

\begin{subequations}
\begin{align}
  V_s &= \underbrace{\frac{d_{33}}{\epsilon^T s^D n}}_{g_s} d_L \label{eq:test_apa_piezo_strain_to_voltage} \\
  F_a &= \underbrace{d_{33} n k_a}_{g_a} \cdot V_a, \quad k_a = \frac{c^{E} A}{L} \label{eq:test_apa_piezo_voltage_to_force}
\end{align}
\end{subequations}

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:stack_parameters}), the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
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{Experimental Validation}

goal: validation of the procedure.

\begin{itemize}
\item Explain test bench: (Figure \ref{fig:detail_fem_apa95ml_bench})
\begin{itemize}
\item 5.7kg granite, vertical guided with an air bearing
\item fibered interferometer measured the vertical motion of the granite \(y\)
\item DAC generating control signal \(u\), voltage amplifier gain of 20, \(V_a\) is the voltage across the two piezoelectric stacks
\item ADC is used to measured the voltage across the piezoelectric sensor stack
\end{itemize}
\end{itemize}

\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}

\begin{itemize}
\item Explain how to experimentally measure the transfer function:
\begin{itemize}
\item test signal, here noise
\item compute and show the transfer functions from \(V_a\) to \(y\) and to \(V_s\)
\item Compare the model and measurement: validation (Figure \ref{fig:detail_fem_apa95ml_comp_plant})
\item talk about the phase:
\begin{itemize}
\item for force sensor, just delay linked to the limited sampling rate of \(0.1\,ms\)
\item for interferometer: additional delay due to electronics being used
\end{itemize}
\item good match. The gains can be further tuned based on the experimental results.
\item[{$\square$}] talk about minimum phase zero: will be discussed during the experimental phase
\end{itemize}
\end{itemize}

\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 $d_i$}
\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 \(d_i\) (\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}

goal:
\begin{itemize}
\item validate the use of super element for control tasks
\end{itemize}

The controller used in the Integral Force Feedback Architecture is \eqref{eq:detail_fem_iff_controller}, wtih \(g\) a gain that can be tuned.

\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}


Above 2 Hz the controller is basically an integrator, whereas an high pass filter is added at 0.5Hz to further reduce the low frequency gain.
In the frequency band of interest, this controller should mostly act as a pure integrator.

\begin{itemize}
\item[{$\square$}] Maybe make a block diagram of the control with added damped input
\end{itemize}

\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.}
\end{figure}

\section*{Conclusion}
\begin{itemize}
\item Validation of the method
\item Very useful to optimize different parts
\item However, model order may become very large and not convenient to perform time domain simulations
\item But extracting dynamics is not computational intensive, even for large model orders
\item For instance APA: order 48, 6 APA for the nano hexapod 288 orders just for the APA

\item[{$\square$}] \href{file:///home/thomas/Cloud/research/papers/published/brumund21\_multib\_simul\_reduc\_order\_flexib\_bodies\_fea/paper/brumund21\_multib\_simul\_reduc\_order\_flexib\_bodies\_fea.pdf}{published paper}
\end{itemize}

\chapter{Actuator}
\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{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{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{Identification of the APA Characteristics}
A first validation of the FEM and inclusion of the ``reduced order flexible model'' in the multi body-model is performed by computed some key characteristics of the APA that can be compared against the datasheet.

\paragraph{Stiffness}
The stiffness is estimated by extracting the transfer function from a vertical force applied on the top frame to the displacement of the same top frame.
The inverse of the DC gain this transfer function should be equal to the axial stiffness of the APA300ML.
A value of \(1.75\,N/\mu m\) is found which is close to the specified stiffness in the datasheet of \(k = 1.8\,N/\mu m\).
See compliance transfer function \ref{fig:detail_fem_apa300ml_compliance}.

\paragraph{Resonance Frequency}

The resonance frequency in the block-free condition is specified to be between 650Hz and 840Hz.
This is estimated at 709Hz from the model (Figure \ref{fig:detail_fem_apa300ml_compliance}).

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_fem_apa300ml_compliance.png}
\caption{\label{fig:detail_fem_apa300ml_compliance}Estimated compliance of the APA300ML}
\end{figure}


\paragraph{Amplification Factor and Actuator Stroke}

The amplification factor is the ratio of the vertical displacement to the (horizontal) stack displacement.
It can be estimated from the multi-body model by computing the transfer function from the horizontal motion of the stacks to the vertical motion of the APA.
The ratio between the two is found to be equal to \(5\).
This is linked to the

From the data-sheet of the piezoelectric stacks (see Table \ref{tab:detail_fem_stack_parameters}, page \pageref{tab:detail_fem_stack_parameters}), the nominal stroke of the stack is \(20\,\mu m\) (which is typical for PZT to have a maximum stroke equal to \(0.1\,\%\) of its length, here equal to \(20\,mm\)).
Three stacks are used, for an horizontal stroke of the stacks of \(60\,\mu m\).
With an amplification factor equal to \(5\), the vertical stroke is estimated at \(300\,\mu m\), which corresponds to what is indicated in the datasheet.


This analysis provides some confidence on the model accuracy.

\section{Simpler 2DoF Model of the APA300ML}
\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}

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}
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}

\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}
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: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}

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: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: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}

\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}

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:detail_fem_conclusion}

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