724 lines
30 KiB
TeX
724 lines
30 KiB
TeX
% Created 2024-03-19 Tue 11:08
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% Intended LaTeX compiler: pdflatex
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\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
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\input{preamble.tex}
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\bibliography{nass-fem.bib}
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\author{Dehaeze Thomas}
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\date{\today}
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\title{NASS - Finite Element Models with Simscape}
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\hypersetup{
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pdfauthor={Dehaeze Thomas},
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pdftitle={NASS - Finite Element Models with Simscape},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 29.2 (Org mode 9.7)},
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pdflang={English}}
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\usepackage{biblatex}
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\begin{document}
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\maketitle
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\tableofcontents
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\clearpage
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In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis.
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It is divided in the following sections:
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\begin{itemize}
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\item Section \ref{sec:APA300ML}:
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A super-element of the Amplified Piezoelectric Actuator APA300ML used for the NASS is exported using Ansys and imported in Simscape.
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The static and dynamical properties of the APA300ML are then estimated using the Simscape model.
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\item Section \ref{sec:flexor_ID16}:
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A first geometry of a Flexible joint is modelled and its characteristics are identified from the Stiffness matrix as well as from the Simscape model.
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\item Section \ref{sec:flexor_025}:
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An optimized flexible joint is developed for the Nano-Hexapod and is then imported in a Simscape model.
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\item Section \ref{sec:strut_encoder}:
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A super element of a complete strut is studied.
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\end{itemize}
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\chapter{APA300ML}
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\label{sec:APA300ML}
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In this section, the Amplified Piezoelectric Actuator APA300ML (\href{doc/APA300ML.pdf}{doc}) is modeled using a Finite Element Software.
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Then a \emph{super element} is exported and imported in Simscape where its dynamic is studied.
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A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure \ref{fig:apa300ml_ansys}.
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The remote point used are also shown in this figure.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=0.7\linewidth]{figs/apa300ml_ansys.jpg}
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\caption{\label{fig:apa300ml_ansys}Ansys FEM of the APA300ML}
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\end{figure}
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\section{Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates}
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We first extract the stiffness and mass matrices.
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\begin{table}[htbp]
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\caption{\label{tab:APA300ML_mat_K}First 10x10 elements of the Stiffness matrix}
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\centering
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\tiny
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\begin{tabularx}{\linewidth}{XXXXXXXXXX}
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\toprule
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200000000 & 30000 & -20000 & -70 & 300000 & 40 & 10000000 & 10000 & -6000 & 30\\
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30000 & 30000000 & 2000 & -200000 & 60 & -10 & 4000 & 2000000 & -500 & 9000\\
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-20000 & 2000 & 7000000 & -10 & -30 & 10 & 6000 & 900 & -500000 & 3\\
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-70 & -200000 & -10 & 1000 & -0.1 & 08 & -20 & -9000 & 3 & -30\\
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300000 & 60 & -30 & -0.1 & 900 & 0.1 & 30000 & 20 & -10 & 06\\
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40 & -10 & 10 & 08 & 0.1 & 10000 & 20 & 9 & -5 & 03\\
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10000000 & 4000 & 6000 & -20 & 30000 & 20 & 200000000 & 10000 & 9000 & 50\\
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10000 & 2000000 & 900 & -9000 & 20 & 9 & 10000 & 30000000 & -500 & 200000\\
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-6000 & -500 & -500000 & 3 & -10 & -5 & 9000 & -500 & 7000000 & -2\\
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30 & 9000 & 3 & -30 & 06 & 03 & 50 & 200000 & -2 & 1000\\
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\bottomrule
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\end{tabularx}
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\end{table}
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\begin{table}[htbp]
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\caption{\label{tab:APA300ML_mat_M}First 10x10 elements of the Mass matrix}
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\centering
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\tiny
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\begin{tabularx}{\linewidth}{XXXXXXXXXX}
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\toprule
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0.01 & -2e-06 & 1e-06 & 6e-09 & 5e-05 & -5e-09 & -0.0005 & -7e-07 & 6e-07 & -3e-09\\
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-2e-06 & 0.01 & 8e-07 & -2e-05 & -8e-09 & 2e-09 & -9e-07 & -0.0002 & 1e-08 & -9e-07\\
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1e-06 & 8e-07 & 0.009 & 5e-10 & 1e-09 & -1e-09 & -5e-07 & 3e-08 & 6e-05 & 1e-10\\
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6e-09 & -2e-05 & 5e-10 & 3e-07 & 2e-11 & -3e-12 & 3e-09 & 9e-07 & -4e-10 & 3e-09\\
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5e-05 & -8e-09 & 1e-09 & 2e-11 & 6e-07 & -4e-11 & -1e-06 & -2e-09 & 1e-09 & -8e-12\\
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-5e-09 & 2e-09 & -1e-09 & -3e-12 & -4e-11 & 1e-07 & -2e-09 & -1e-09 & -4e-10 & -5e-12\\
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-0.0005 & -9e-07 & -5e-07 & 3e-09 & -1e-06 & -2e-09 & 0.01 & 1e-07 & -3e-07 & -2e-08\\
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-7e-07 & -0.0002 & 3e-08 & 9e-07 & -2e-09 & -1e-09 & 1e-07 & 0.01 & -4e-07 & 2e-05\\
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6e-07 & 1e-08 & 6e-05 & -4e-10 & 1e-09 & -4e-10 & -3e-07 & -4e-07 & 0.009 & -2e-10\\
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-3e-09 & -9e-07 & 1e-10 & 3e-09 & -8e-12 & -5e-12 & -2e-08 & 2e-05 & -2e-10 & 3e-07\\
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\bottomrule
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\end{tabularx}
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\end{table}
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Then, we extract the coordinates of the interface nodes.
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\begin{table}[htbp]
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\caption{\label{tab:coordinate_nodes}Coordinates of the interface nodes}
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\centering
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\begin{tabularx}{0.6\linewidth}{ccccc}
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\toprule
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Node i & Node Number & x [m] & y [m] & z [m]\\
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\midrule
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1.0 & 697783 & 0.0 & 0.0 & -0.015\\
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2.0 & 697784 & 0.0 & 0.0 & 0.015\\
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3.0 & 697785 & -0.0325 & 0.0 & 0.0\\
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4.0 & 697786 & -0.0125 & 0.0 & 0.0\\
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5.0 & 697787 & -0.0075 & 0.0 & 0.0\\
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6.0 & 697788 & 0.0125 & 0.0 & 0.0\\
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7.0 & 697789 & 0.0325 & 0.0 & 0.0\\
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\bottomrule
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\end{tabularx}
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\end{table}
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\begin{table}[htbp]
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\caption{\label{tab:parameters_fem}Some extracted parameters of the FEM}
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\centering
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\begin{tabularx}{0.4\linewidth}{lc}
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\toprule
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Total number of Nodes & 7\\
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Number of interface Nodes & 7\\
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Number of Modes & 120\\
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Size of M and K matrices & 162\\
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\bottomrule
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\end{tabularx}
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\end{table}
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Using \texttt{K}, \texttt{M} and \texttt{int\_xyz}, we can now use the \texttt{Reduced Order Flexible Solid} simscape block.
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\section{Piezoelectric parameters}
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In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
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PZT-4
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The ratio of the developed force to applied voltage is:
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\begin{equation}
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\label{eq:piezo_voltage_to_force}
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F_a = g_a V_a, \quad g_a = d_{33} n k_a
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\end{equation}
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where:
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\begin{itemize}
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\item \(F_a\): developed force in [N]
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\item \(n\): number of layers of the actuator stack
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\item \(d_{33}\): strain constant in [m/V]
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\item \(k_a\): actuator stack stiffness in [N/m]
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\item \(V_a\): applied voltage in [V]
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\end{itemize}
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If we take the numerical values, we obtain:
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From \cite{fleming14_desig_model_contr_nanop_system} (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
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\begin{equation}
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\label{eq:piezo_strain_to_voltage}
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V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
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\end{equation}
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where:
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\begin{itemize}
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\item \(V_s\): measured voltage in [V]
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\item \(d_{33}\): strain constant in [m/V]
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\item \(\epsilon^T\): permittivity under constant stress in [F/m]
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\item \(s^D\): elastic compliance under constant electric displacement in [m\^{}2/N]
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\item \(n\): number of layers of the sensor stack
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\item \(\Delta h\): relative displacement in [m]
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\end{itemize}
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If we take the numerical values, we obtain:
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\section{Simscape Model}
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The flexible element is imported using the \texttt{Reduced Order Flexible Solid} simscape block.
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Let's say we use two stacks as a force sensor and one stack as an actuator:
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\begin{itemize}
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\item A \texttt{Relative Motion Sensor} block is added between the nodes A and C
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\item An \texttt{Internal Force} block is added between the remote points E and B
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\end{itemize}
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The interface nodes are shown in Figure \ref{fig:apa300ml_ansys}.
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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).
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\section{Identification of the APA Characteristics}
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\subsection{Stiffness}
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The transfer function from vertical external force to the relative vertical displacement is identified.
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The inverse of its DC gain is the axial stiffness of the APA:
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The specified stiffness in the datasheet is \(k = 1.8\, [N/\mu m]\).
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\subsection{Resonance Frequency}
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The resonance frequency is specified to be between 650Hz and 840Hz.
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This is also the case for the FEM model (Figure \ref{fig:apa300ml_resonance}).
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/apa300ml_resonance.png}
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\caption{\label{fig:apa300ml_resonance}First resonance is around 800Hz}
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\end{figure}
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\subsection{Amplification factor}
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The amplification factor is the ratio of the vertical displacement to the stack displacement.
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The ratio of the two displacement is computed from the FEM model.
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This is actually correct and approximately corresponds to the ratio of the piezo height and length:
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\subsection{Stroke}
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Estimation of the actuator stroke:
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\[ \Delta H = A n \Delta L \]
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with:
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\begin{itemize}
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\item \(\Delta H\) Axial Stroke of the APA
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\item \(A\) Amplification factor (5 for the APA300ML)
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\item \(n\) Number of stack used
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\item \(\Delta L\) Stroke of the stack (0.1\% of its length)
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\end{itemize}
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This is exactly the specified stroke in the data-sheet.
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\subsection{Stroke BIS}
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\begin{itemize}
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\item[{$\square$}] Identified the stroke form the transfer function from V to z
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\end{itemize}
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\section{Identification of the Dynamics from actuator to replace displacement}
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We first set the mass to be approximately zero.
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The dynamics is identified from the applied force to the measured relative displacement.
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The same dynamics is identified for a payload mass of 10Kg.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/apa300ml_plant_dynamics.png}
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\caption{\label{fig:apa300ml_plant_dynamics}Transfer function from forces applied by the stack to the axial displacement of the APA}
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\end{figure}
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The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure \ref{fig:apa300ml_dvf_root_locus}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/apa300ml_dvf_root_locus.png}
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\caption{\label{fig:apa300ml_dvf_root_locus}Root Locus for Direct Velocity Feedback}
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\end{figure}
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\section{Identification of the Dynamics from actuator to force sensor}
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Let's use 2 stacks as a force sensor and 1 stack as force actuator.
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The transfer function from actuator voltage to sensor voltage is identified and shown in Figure \ref{fig:apa300ml_iff_plant}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/apa300ml_iff_plant.png}
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\caption{\label{fig:apa300ml_iff_plant}Transfer function from actuator to force sensor}
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\end{figure}
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For root locus corresponding to IFF is shown in Figure \ref{fig:apa300ml_iff_root_locus}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/apa300ml_iff_root_locus.png}
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\caption{\label{fig:apa300ml_iff_root_locus}Root Locus for IFF}
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\end{figure}
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\section{Identification for a simpler model}
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The goal in this section is to identify the parameters of a simple APA model from the FEM.
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This can be useful is a lower order model is to be used for simulations.
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The presented model is based on \cite{souleille18_concep_activ_mount_space_applic}.
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The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure \ref{fig:souleille18_model_piezo}).
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The parameters are shown in the table below.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=0.7\linewidth]{./figs/souleille18_model_piezo.png}
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\caption{\label{fig:souleille18_model_piezo}Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator}
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\end{figure}
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\begin{table}[htbp]
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\caption{\label{tab:parameters_apa100m}Parameters used for the model of the APA 100M}
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\centering
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\begin{tabularx}{0.8\linewidth}{cl}
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\toprule
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& Meaning\\
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\midrule
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\(k_e\) & Stiffness used to adjust the pole of the isolator\\
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\(k_1\) & Stiffness of the metallic suspension when the stack is removed\\
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\(k_a\) & Stiffness of the actuator\\
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\(c_1\) & Added viscous damping\\
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\bottomrule
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\end{tabularx}
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\end{table}
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The goal is to determine \(k_e\), \(k_a\) and \(k_1\) so that the simplified model fits the FEM model.
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\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
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\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
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If we can fix \(k_a\), we can determine \(k_e\) and \(k_1\) with:
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\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
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\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
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From the identified dynamics, compute \(\alpha\) and \(\beta\)
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\(k_a\) is estimated using the following formula:
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The factor can be adjusted to better match the curves.
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Then \(k_e\) and \(k_1\) are computed.
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\begin{table}[htbp]
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\caption{\label{tab:stiffnesses_apa100m}Obtained stiffnesses of the APA 100M}
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\centering
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\begin{tabularx}{0.3\linewidth}{cl}
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\toprule
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& Value [N/um]\\
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\midrule
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ka & 40.5\\
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ke & 1.5\\
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k1 & 0.4\\
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\bottomrule
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\end{tabularx}
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\end{table}
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The damping in the system is adjusted to match the FEM model if necessary.
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The analytical model of the simpler system is defined below:
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And the DC gain is adjusted for the force sensor:
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The dynamics of the FEM model and the simpler model are compared in Figure \ref{fig:apa300ml_comp_simpler_model}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=1.0\linewidth]{figs/apa300ml_comp_simpler_model.png}
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\caption{\label{fig:apa300ml_comp_simpler_model}Comparison of the Dynamics between the FEM model and the simplified one}
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\end{figure}
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The simplified model has also been implemented in Simscape.
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The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure \ref{fig:apa300ml_comp_simpler_simscape}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/apa300ml_comp_simpler_simscape.png}
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\caption{\label{fig:apa300ml_comp_simpler_simscape}Comparison of the Dynamics between the FEM model and the simplified simscape model}
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\end{figure}
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\section{Integral Force Feedback}
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In this section, Integral Force Feedback control architecture is applied on the APA300ML.
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First, the plant (dynamics from voltage actuator to voltage sensor is identified).
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The payload mass is set to 10kg.
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The obtained dynamics is shown in Figure \ref{fig:piezo_amplified_iff_plant}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/piezo_amplified_iff_plant.png}
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\caption{\label{fig:piezo_amplified_iff_plant}IFF Plant}
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\end{figure}
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The controller is defined below and the loop gain is shown in Figure \ref{fig:piezo_amplified_iff_loop_gain}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/piezo_amplified_iff_loop_gain.png}
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\caption{\label{fig:piezo_amplified_iff_loop_gain}IFF Loop Gain}
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\end{figure}
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Now the closed-loop system is identified again and compare with the open loop system in Figure \ref{fig:piezo_amplified_iff_comp}.
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It is the expected behavior as shown in the Figure \ref{fig:souleille18_results} (from \cite{souleille18_concep_activ_mount_space_applic}).
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/piezo_amplified_iff_comp.png}
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\caption{\label{fig:piezo_amplified_iff_comp}OL and CL transfer functions}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=1.0\linewidth]{figs/souleille18_results.png}
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\caption{\label{fig:souleille18_results}Results obtained in \cite{souleille18_concep_activ_mount_space_applic}}
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\end{figure}
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\chapter{First Flexible Joint Geometry}
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\label{sec:flexor_ID16}
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The studied flexor is shown in Figure \ref{fig:flexor_id16_screenshot}.
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The stiffness and mass matrices representing the dynamics of the flexor are exported from a FEM.
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It is then imported into Simscape.
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A simplified model of the flexor is then developped.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=0.3\linewidth]{figs/flexor_id16_screenshot.png}
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\caption{\label{fig:flexor_id16_screenshot}Flexor studied}
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\end{figure}
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\section{Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates}
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We first extract the stiffness and mass matrices.
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\begin{table}[htbp]
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\caption{\label{tab:APA300ML_mat_K_2MDoF}First 10x10 elements of the Stiffness matrix}
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\centering
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\tiny
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\begin{tabularx}{\linewidth}{XXXXXXXXXX}
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\toprule
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11200000 & 195 & 2220 & -0.719 & -265 & 1.59 & -11200000 & -213 & -2220 & 0.147 & & 195 & 11400000 & 1290 & -148 & -0.188 & 2.41 & -212 & -11400000 & -1290 & 148\\
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2220 & 1290 & 119000000 & 1.31 & 1.49 & 1.79 & -2220 & -1290 & -119000000 & -1.31 & & & & & & & & & & & \\
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-0.719 & -148 & 1.31 & 33 & 000488 & -000977 & 0.141 & 148 & -1.31 & -33 & & & & & & & & & & & \\
|
|
-265 & -0.188 & 1.49 & 000488 & 33 & 00293 & 266 & 0.154 & -1.49 & 00026 & & & & & & & & & & & \\
|
|
1.59 & 2.41 & 1.79 & -000977 & 00293 & 236 & -1.32 & -2.55 & -1.79 & 000379 & & & & & & & & & & & \\
|
|
-11200000 & -212 & -2220 & 0.141 & 266 & -1.32 & 11400000 & 24600 & 1640 & 120 & & & & & & & & & & & \\
|
|
-213 & -11400000 & -1290 & 148 & 0.154 & -2.55 & 24600 & 11400000 & 1290 & -72 & & & & & & & & & & & \\
|
|
-2220 & -1290 & -119000000 & -1.31 & -1.49 & -1.79 & 1640 & 1290 & 119000000 & 1.32 & & & & & & & & & & & \\
|
|
0.147 & 148 & -1.31 & -33 & 00026 & 000379 & 120 & -72 & 1.32 & 34.7 & & & & & & & & & & & \\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:APA300ML_mat_M_2MDoF}First 10x10 elements of the Mass matrix}
|
|
\centering
|
|
\tiny
|
|
\begin{tabularx}{\linewidth}{XXXXXXXXXX}
|
|
\toprule
|
|
0.02 & 1e-09 & -4e-08 & -1e-10 & 0.0002 & -3e-11 & 0.004 & 5e-08 & 7e-08 & 1e-10\\
|
|
1e-09 & 0.02 & -3e-07 & -0.0002 & -1e-10 & -2e-09 & 2e-08 & 0.004 & 3e-07 & 1e-05\\
|
|
-4e-08 & -3e-07 & 0.02 & 7e-10 & -2e-09 & 1e-09 & 3e-07 & 7e-08 & 0.003 & 1e-09\\
|
|
-1e-10 & -0.0002 & 7e-10 & 4e-06 & -1e-12 & -6e-13 & 2e-10 & -7e-06 & -8e-10 & -1e-09\\
|
|
0.0002 & -1e-10 & -2e-09 & -1e-12 & 3e-06 & 2e-13 & 9e-06 & 4e-11 & 2e-09 & -3e-13\\
|
|
-3e-11 & -2e-09 & 1e-09 & -6e-13 & 2e-13 & 4e-07 & 8e-11 & 9e-10 & -1e-09 & 2e-12\\
|
|
0.004 & 2e-08 & 3e-07 & 2e-10 & 9e-06 & 8e-11 & 0.02 & -7e-08 & -3e-07 & -2e-10\\
|
|
5e-08 & 0.004 & 7e-08 & -7e-06 & 4e-11 & 9e-10 & -7e-08 & 0.01 & -4e-08 & 0.0002\\
|
|
7e-08 & 3e-07 & 0.003 & -8e-10 & 2e-09 & -1e-09 & -3e-07 & -4e-08 & 0.02 & -1e-09\\
|
|
1e-10 & 1e-05 & 1e-09 & -1e-09 & -3e-13 & 2e-12 & -2e-10 & 0.0002 & -1e-09 & 2e-06\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:parameters_fem_2MDoF}Some extracted parameters of the FEM}
|
|
\centering
|
|
\begin{tabularx}{0.4\linewidth}{lc}
|
|
\toprule
|
|
Total number of Nodes & 2\\
|
|
Number of interface Nodes & 2\\
|
|
Number of Modes & 6\\
|
|
Size of M and K matrices & 18\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
Then, we extract the coordinates of the interface nodes.
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:coordinate_nodes_2MDoF}Coordinates of the interface nodes}
|
|
\centering
|
|
\begin{tabularx}{0.6\linewidth}{ccccc}
|
|
\toprule
|
|
Node i & Node Number & x [m] & y [m] & z [m]\\
|
|
\midrule
|
|
1.0 & 181278.0 & 0.0 & 0.0 & 0.0\\
|
|
2.0 & 181279.0 & 0.0 & 0.0 & -0.0\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
Using \texttt{K}, \texttt{M} and \texttt{int\_xyz}, we can use the \texttt{Reduced Order Flexible Solid} simscape block.
|
|
\section{Identification of the parameters using Simscape and looking at the Stiffness Matrix}
|
|
The flexor is now imported into Simscape and its parameters are estimated using an identification.
|
|
|
|
The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
|
|
And we find the same parameters as the one estimated from the Stiffness matrix.
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:comp_identified_stiffnesses}Comparison of identified and FEM stiffnesses}
|
|
\centering
|
|
\begin{tabularx}{0.7\linewidth}{lcc}
|
|
\toprule
|
|
\textbf{Characteristic} & \textbf{Value} & \textbf{Identification}\\
|
|
\midrule
|
|
Axial Stiffness Dz [N/um] & 119 & 119\\
|
|
Bending Stiffness Rx [Nm/rad] & 33 & 33\\
|
|
Bending Stiffness Ry [Nm/rad] & 33 & 33\\
|
|
Torsion Stiffness Rz [Nm/rad] & 236 & 236\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
\section{Simpler Model}
|
|
Let's now model the flexible joint with a ``perfect'' Bushing joint as shown in Figure \ref{fig:flexible_joint_simscape}.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1,width=0.3\linewidth]{figs/flexible_joint_simscape.png}
|
|
\caption{\label{fig:flexible_joint_simscape}Bushing Joint used to model the flexible joint}
|
|
\end{figure}
|
|
|
|
The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
|
|
The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
|
|
The two obtained dynamics are compared in Figure
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/flexor_ID16_compare_bushing_joint.png}
|
|
\caption{\label{fig:flexor_ID16_compare_bushing_joint}Comparison of the Joint compliance between the FEM model and the simpler model}
|
|
\end{figure}
|
|
\chapter{Optimized Flexible Joint}
|
|
\label{sec:flexor_025}
|
|
The joint geometry has been optimized using Ansys to have lower bending stiffness while keeping a large axial stiffness.
|
|
|
|
The obtained geometry is shown in Figure \ref{fig:optimal_flexor}.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1,width=1.0\linewidth]{figs/flexor_025_MDoF.jpg}
|
|
\caption{\label{fig:optimal_flexor}Flexor studied}
|
|
\end{figure}
|
|
\section{Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates}
|
|
We first extract the stiffness and mass matrices.
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:flex_mat_K}First 10x10 elements of the Stiffness matrix}
|
|
\centering
|
|
\tiny
|
|
\begin{tabularx}{\linewidth}{XXXXXXXXXX}
|
|
\toprule
|
|
12700000 & -18.5 & -26.8 & 00162 & -4.63 & 64 & -12700000 & 18.3 & 26.7 & 00234\\
|
|
-18.5 & 12700000 & -499 & -132 & 00414 & -0.495 & 18.4 & -12700000 & 499 & 132\\
|
|
-26.8 & -499 & 94000000 & -470 & 00771 & -0.855 & 26.8 & 498 & -94000000 & 470\\
|
|
00162 & -132 & -470 & 4.83 & 2.61e-07 & 000123 & -00163 & 132 & 470 & -4.83\\
|
|
-4.63 & 00414 & 00771 & 2.61e-07 & 4.83 & 4.43e-05 & 4.63 & -00413 & -00772 & -4.3e-07\\
|
|
64 & -0.495 & -0.855 & 000123 & 4.43e-05 & 260 & -64 & 0.495 & 0.855 & -000124\\
|
|
-12700000 & 18.4 & 26.8 & -00163 & 4.63 & -64 & 12700000 & -18.2 & -26.7 & -00234\\
|
|
18.3 & -12700000 & 498 & 132 & -00413 & 0.495 & -18.2 & 12700000 & -498 & -132\\
|
|
26.7 & 499 & -94000000 & 470 & -00772 & 0.855 & -26.7 & -498 & 94000000 & -470\\
|
|
00234 & 132 & 470 & -4.83 & -4.3e-07 & -000124 & -00234 & -132 & -470 & 4.83\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:flex_mat_M}First 10x10 elements of the Mass matrix}
|
|
\centering
|
|
\tiny
|
|
\begin{tabularx}{\linewidth}{XXXXXXXXXX}
|
|
\toprule
|
|
0.006 & 8e-09 & -2e-08 & -1e-10 & 3e-05 & 3e-08 & 0.003 & -3e-09 & 9e-09 & 2e-12\\
|
|
8e-09 & 0.02 & 1e-07 & -3e-05 & 1e-11 & 6e-10 & 1e-08 & 0.003 & -5e-08 & 3e-09\\
|
|
-2e-08 & 1e-07 & 0.01 & -6e-08 & -6e-11 & -8e-12 & -1e-07 & 1e-08 & 0.003 & -1e-08\\
|
|
-1e-10 & -3e-05 & -6e-08 & 1e-06 & 7e-14 & 6e-13 & 1e-10 & 1e-06 & -1e-08 & 3e-10\\
|
|
3e-05 & 1e-11 & -6e-11 & 7e-14 & 2e-07 & 1e-10 & 3e-08 & -7e-12 & 6e-11 & -6e-16\\
|
|
3e-08 & 6e-10 & -8e-12 & 6e-13 & 1e-10 & 5e-07 & 1e-08 & -5e-10 & -1e-11 & 1e-13\\
|
|
0.003 & 1e-08 & -1e-07 & 1e-10 & 3e-08 & 1e-08 & 0.02 & -2e-08 & 1e-07 & -4e-12\\
|
|
-3e-09 & 0.003 & 1e-08 & 1e-06 & -7e-12 & -5e-10 & -2e-08 & 0.006 & -8e-08 & 3e-05\\
|
|
9e-09 & -5e-08 & 0.003 & -1e-08 & 6e-11 & -1e-11 & 1e-07 & -8e-08 & 0.01 & -6e-08\\
|
|
2e-12 & 3e-09 & -1e-08 & 3e-10 & -6e-16 & 1e-13 & -4e-12 & 3e-05 & -6e-08 & 2e-07\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
|
|
Then, we extract the coordinates of the interface nodes.
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:coordinate_nodes_flex}Coordinates of the interface nodes for Flexible Joint}
|
|
\centering
|
|
\begin{tabularx}{0.6\linewidth}{ccccc}
|
|
\toprule
|
|
Total number of Nodes & 2\\
|
|
Number of interface Nodes & 2\\
|
|
Number of Modes & 6\\
|
|
Size of M and K matrices & 18\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:coordinate_interface_nodes_flex}Coordinates of the interface nodes}
|
|
\centering
|
|
\begin{tabularx}{0.6\linewidth}{ccccc}
|
|
\toprule
|
|
Node i & Node Number & x [m] & y [m] & z [m]\\
|
|
\midrule
|
|
1.0 & 528875.0 & 0.0 & 0.0 & 0.0\\
|
|
2.0 & 528876.0 & 0.0 & 0.0 & -0.0\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
Using \texttt{K}, \texttt{M} and \texttt{int\_xyz}, we can use the \texttt{Reduced Order Flexible Solid} simscape block.
|
|
\section{Identification of the parameters using Simscape}
|
|
The flexor is now imported into Simscape and its parameters are estimated using an identification.
|
|
|
|
The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
|
|
And we find the same parameters as the one estimated from the Stiffness matrix.
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:comp_identified_stiffnesses_flex}Comparison of identified and FEM stiffnesses}
|
|
\centering
|
|
\begin{tabularx}{0.7\linewidth}{lcc}
|
|
\toprule
|
|
\textbf{Caracteristic} & \textbf{Value} & \textbf{Identification}\\
|
|
\midrule
|
|
Axial Stiffness Dz [N/um] & 94.0 & 93.9\\
|
|
Bending Stiffness Rx [Nm/rad] & 4.8 & 4.8\\
|
|
Bending Stiffness Ry [Nm/rad] & 4.8 & 4.8\\
|
|
Torsion Stiffness Rz [Nm/rad] & 260.2 & 260.2\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
\section{Simpler Model}
|
|
Let's now model the flexible joint with a ``perfect'' Bushing joint as shown in Figure \ref{fig:flexible_joint_simscape}.
|
|
|
|
The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
|
|
The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
|
|
The two obtained dynamics are compared in Figure
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/flexor_ID16_compare_bushing_joint_optimized.png}
|
|
\caption{\label{fig:flexor_ID16_compare_bushing_joint_optimized}Comparison of the Joint compliance between the FEM model and the simpler model}
|
|
\end{figure}
|
|
\section{Comparison with a stiffer Flexible Joint}
|
|
The stiffness matrix with the flexible joint with a ``hinge'' size of 0.50mm is loaded.
|
|
Its parameters are compared with the Flexible Joint with a size of 0.25mm in the table below.
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:comp_stiffnesses_flex_size}Comparison of flexible joint stiffnesses for 0.25 and 0.5mm}
|
|
\centering
|
|
\begin{tabularx}{0.7\linewidth}{lcc}
|
|
\toprule
|
|
\textbf{Caracteristic} & \textbf{0.25 mm} & \textbf{0.50 mm}\\
|
|
\midrule
|
|
Axial Stiffness Dz [N/um] & 94.0 & 124.7\\
|
|
Shear Stiffness [N/um] & 12.7 & 25.8\\
|
|
Bending Stiffness Rx [Nm/rad] & 4.8 & 26.0\\
|
|
Bending Stiffness Ry [Nm/rad] & 4.8 & 26.0\\
|
|
Torsion Stiffness Rz [Nm/rad] & 260.2 & 538.0\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
\chapter{Complete Strut with Encoder}
|
|
\label{sec:strut_encoder}
|
|
\section{Introduction}
|
|
|
|
Now, the full nano-hexapod strut is modelled using Ansys.
|
|
|
|
The 3D as well as the interface nodes are shown in Figure \ref{fig:strut_encoder_points3}.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1,width=1.0\linewidth]{figs/strut_fem_nodes.jpg}
|
|
\caption{\label{fig:strut_encoder_points3}Interface points}
|
|
\end{figure}
|
|
|
|
A side view is shown in Figure \ref{fig:strut_encoder_nodes_side}.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1,width=1.0\linewidth]{figs/strut_fem_nodes_side.jpg}
|
|
\caption{\label{fig:strut_encoder_nodes_side}Interface points - Side view}
|
|
\end{figure}
|
|
|
|
The flexible joints used have a 0.25mm width size.
|
|
\section{Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates}
|
|
We first extract the stiffness and mass matrices.
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:strut_mat_K}First 10x10 elements of the Stiffness matrix}
|
|
\centering
|
|
\tiny
|
|
\begin{tabularx}{\linewidth}{XXXXXXXXXX}
|
|
\toprule
|
|
2000000 & 1000000 & -3000000 & -400 & 300 & 200 & -30 & 2000 & -10000 & 0.3\\
|
|
1000000 & 4000000 & -8000000 & -900 & 400 & -50 & -6000 & 10000 & -20000 & 3\\
|
|
-3000000 & -8000000 & 20000000 & 2000 & -900 & 200 & -10000 & 20000 & -300000 & 7\\
|
|
-400 & -900 & 2000 & 5 & -0.1 & 05 & 1 & -3 & 6 & -0007\\
|
|
300 & 400 & -900 & -0.1 & 5 & 04 & -0.1 & 0.5 & -3 & 0001\\
|
|
200 & -50 & 200 & 05 & 04 & 300 & 4 & -01 & -1 & 3e-05\\
|
|
-30 & -6000 & -10000 & 1 & -0.1 & 4 & 3000000 & -1000000 & -2000000 & -300\\
|
|
2000 & 10000 & 20000 & -3 & 0.5 & -01 & -1000000 & 6000000 & 7000000 & 1000\\
|
|
-10000 & -20000 & -300000 & 6 & -3 & -1 & -2000000 & 7000000 & 20000000 & 2000\\
|
|
0.3 & 3 & 7 & -0007 & 0001 & 3e-05 & -300 & 1000 & 2000 & 5\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:strut_mat_M}First 10x10 elements of the Mass matrix}
|
|
\centering
|
|
\tiny
|
|
\begin{tabularx}{\linewidth}{XXXXXXXXXX}
|
|
\toprule
|
|
0.04 & -0.005 & 0.007 & 2e-06 & 0.0001 & -5e-07 & -1e-05 & -9e-07 & 8e-05 & -5e-10\\
|
|
-0.005 & 0.03 & 0.02 & -0.0001 & 1e-06 & -3e-07 & 3e-05 & -0.0001 & 8e-05 & -3e-08\\
|
|
0.007 & 0.02 & 0.08 & -6e-06 & -5e-06 & -7e-07 & 4e-05 & -0.0001 & 0.0005 & -3e-08\\
|
|
2e-06 & -0.0001 & -6e-06 & 2e-06 & -4e-10 & 2e-11 & -8e-09 & 3e-08 & -2e-08 & 6e-12\\
|
|
0.0001 & 1e-06 & -5e-06 & -4e-10 & 3e-06 & 2e-10 & -3e-09 & 3e-09 & -7e-09 & 6e-13\\
|
|
-5e-07 & -3e-07 & -7e-07 & 2e-11 & 2e-10 & 5e-07 & -2e-08 & 5e-09 & -5e-09 & 1e-12\\
|
|
-1e-05 & 3e-05 & 4e-05 & -8e-09 & -3e-09 & -2e-08 & 0.04 & 0.004 & 0.003 & 1e-06\\
|
|
-9e-07 & -0.0001 & -0.0001 & 3e-08 & 3e-09 & 5e-09 & 0.004 & 0.02 & -0.02 & 0.0001\\
|
|
8e-05 & 8e-05 & 0.0005 & -2e-08 & -7e-09 & -5e-09 & 0.003 & -0.02 & 0.08 & -5e-06\\
|
|
-5e-10 & -3e-08 & -3e-08 & 6e-12 & 6e-13 & 1e-12 & 1e-06 & 0.0001 & -5e-06 & 2e-06\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
|
|
Then, we extract the coordinates of the interface nodes.
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:parameters_fem_strut}Some extracted parameters of the FEM}
|
|
\centering
|
|
\begin{tabularx}{0.4\linewidth}{lc}
|
|
\toprule
|
|
Total number of Nodes & 8\\
|
|
Number of interface Nodes & 8\\
|
|
Number of Modes & 6\\
|
|
Size of M and K matrices & 54\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:coordinate_nodes_strut}Coordinates of the interface nodes}
|
|
\centering
|
|
\begin{tabularx}{0.6\linewidth}{ccccc}
|
|
\toprule
|
|
Node i & Node Number & x [m] & y [m] & z [m]\\
|
|
\midrule
|
|
1.0 & 504411.0 & 0.0 & 0.0 & 0.0405\\
|
|
2.0 & 504412.0 & 0.0 & 0.0 & -0.0405\\
|
|
3.0 & 504413.0 & -0.0325 & 0.0 & 0.0\\
|
|
4.0 & 504414.0 & -0.0125 & 0.0 & 0.0\\
|
|
5.0 & 504415.0 & -0.0075 & 0.0 & 0.0\\
|
|
6.0 & 504416.0 & 0.0325 & 0.0 & 0.0\\
|
|
7.0 & 504417.0 & 0.004 & 0.0145 & -0.00175\\
|
|
8.0 & 504418.0 & 0.004 & 0.0166 & -0.00175\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
Using \texttt{K}, \texttt{M} and \texttt{int\_xyz}, we can use the \texttt{Reduced Order Flexible Solid} simscape block.
|
|
\section{Piezoelectric parameters}
|
|
Parameters for the APA300ML:
|
|
\section{Identification of the Dynamics}
|
|
The dynamics is identified from the applied force to the measured relative displacement.
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The same dynamics is identified for a payload mass of 10Kg.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/dynamics_encoder_full_strut.png}
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\caption{\label{fig:dynamics_encoder_full_strut}Dynamics from the force actuator to the measured motion by the encoder}
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\end{figure}
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\printbibliography[heading=bibintoc,title={Bibliography}]
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\end{document}
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