246 lines
12 KiB
TeX
246 lines
12 KiB
TeX
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% Created 2024-03-19 Tue 11:11
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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-flexible-joints.bib}
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\author{Dehaeze Thomas}
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\date{\today}
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\title{Nano-Hexapod - Flexible Joints Optimization}
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\hypersetup{
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pdfauthor={Dehaeze Thomas},
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pdftitle={Nano-Hexapod - Flexible Joints Optimization},
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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 is studied the effect of the mechanical behavior of the flexible joints that are located the extremities of each nano-hexapod's legs.
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Ideally, we want the x and y rotations to be free and all the translations to be blocked.
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However, this is never the case and be have to consider:
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\begin{itemize}
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\item Finite bending stiffnesses (Section \ref{sec:rot_stiffness})
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\item Axial stiffness in the direction of the legs (Section \ref{sec:trans_stiffness})
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\end{itemize}
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This may impose some limitations, also, the goal is to specify the required joints stiffnesses (summarized in Section \ref{sec:conclusion}).
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\begin{table}[htbp]
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\caption{\label{tab:nass_flexible_joints_section_matlab_code}Report sections and corresponding Matlab files}
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\centering
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\begin{tabularx}{0.6\linewidth}{lX}
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\toprule
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\textbf{Sections} & \textbf{Matlab File}\\
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\midrule
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Section \ref{sec:nass_flexible_joints_1_a} & \texttt{nass\_flexible\_joints\_1\_.m}\\
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\bottomrule
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\end{tabularx}
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\end{table}
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\chapter{Bending and Torsional Stiffness}
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\label{sec:rot_stiffness}
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In this section, we wish to study the effect of the rotation flexibility of the nano-hexapod joints.
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\section{Initialization}
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Let's initialize all the stages with default parameters.
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Let's consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
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\section{Realistic Bending Stiffness Values}
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Let's compare the ideal case (zero stiffness in rotation and infinite stiffness in translation) with a more realistic case:
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\begin{itemize}
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\item \(K_{\theta, \phi} = 15\,[Nm/rad]\) stiffness in flexion
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\item \(K_{\psi} = 20\,[Nm/rad]\) stiffness in torsion
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\end{itemize}
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The stiffness and damping of the nano-hexapod's legs are:
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This corresponds to the optimal identified stiffness.
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\subsection{Direct Velocity Feedback}
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We identify the dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness.
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The obtained dynamics are shown in Figure \ref{fig:flex_joint_rot_dvf}.
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It is shown that the adding of stiffness for the flexible joints does increase a little bit the frequencies of the mass suspension modes. It stiffen the structure.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/flex_joint_rot_dvf.png}
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\caption{\label{fig:flex_joint_rot_dvf}Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint stiffness}
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\end{figure}
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\subsection{Primary Plant}
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Let's now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs).
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The dynamics is compare with and without the joint flexibility in Figure \ref{fig:flex_joints_rot_primary_plant_L}.
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The plant dynamics is not found to be changing significantly.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/flex_joints_rot_primary_plant_L.png}
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\caption{\label{fig:flex_joints_rot_primary_plant_L}Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with perfect joints and with flexible joints}
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\end{figure}
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\subsection{Conclusion}
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\begin{important}
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Considering realistic flexible joint bending stiffness for the nano-hexapod does not seems to impose any limitation on the DVF control nor on the primary control.
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It only increases a little bit the suspension modes of the sample on top of the nano-hexapod.
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\end{important}
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\section{Parametric Study}
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We wish now to see what is the impact of the rotation stiffness of the flexible joints on the dynamics.
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This will help to determine the requirements on the joint's stiffness.
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Let's consider the following bending stiffness of the flexible joints:
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We also consider here a nano-hexapod with the identified optimal actuator stiffness.
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\subsection{Direct Velocity Feedback}
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The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure \ref{fig:flex_joints_rot_study_dvf}.
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The corresponding Root Locus plot is shown in Figure \ref{fig:flex_joints_rot_study_dvf_root_locus}.
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It is shown that the bending stiffness of the flexible joints does indeed change a little the dynamics, but critical damping is stiff achievable with Direct Velocity Feedback.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/flex_joints_rot_study_dvf.png}
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\caption{\label{fig:flex_joints_rot_study_dvf}Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered Rotation Stiffnesses}
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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]{figs/flex_joints_rot_study_dvf_root_locus.png}
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\caption{\label{fig:flex_joints_rot_study_dvf_root_locus}Root Locus for all the considered Rotation Stiffnesses}
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\end{figure}
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\subsection{Primary Control}
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The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure \ref{fig:flex_joints_rot_study_primary_plant}.
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It is shown that the bending stiffness of the flexible joints have very little impact on the dynamics.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/flex_joints_rot_study_primary_plant.png}
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\caption{\label{fig:flex_joints_rot_study_primary_plant}Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered bending stiffnesses}
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\end{figure}
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\subsection{Conclusion}
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\begin{important}
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The bending stiffness of the flexible joint does not significantly change the dynamics.
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\end{important}
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\chapter{Axial Stiffness}
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\label{sec:trans_stiffness}
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Let's know consider a flexibility in translation of the flexible joint, in the axis of the legs.
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\section{Realistic Translation Stiffness Values}
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We choose realistic values of the axial stiffness of the joints:
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\[ K_a = 60\,[N/\mu m] \]
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\subsection{Initialization}
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Let's initialize all the stages with default parameters.
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Let's consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
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\subsection{Direct Velocity Feedback}
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The dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness are identified.
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The obtained dynamics are shown in Figure \ref{fig:flex_joint_trans_dvf}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/flex_joint_trans_dvf.png}
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\caption{\label{fig:flex_joint_trans_dvf}Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint axis stiffness}
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\end{figure}
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\subsection{Primary Plant}
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Let's now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs).
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The dynamics is compare with and without the joint flexibility in Figure \ref{fig:flex_joints_trans_primary_plant_L}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/flex_joints_trans_primary_plant_L.png}
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\caption{\label{fig:flex_joints_trans_primary_plant_L}Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with infinite axis stiffnes (solid) and with realistic axial stiffness (dashed)}
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\end{figure}
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\subsection{Conclusion}
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\begin{important}
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For the realistic value of the flexible joint axial stiffness, the dynamics is not much impact, and this should not be a problem for control.
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\end{important}
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\section{Parametric study}
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We wish now to see what is the impact of the \textbf{axial} stiffness of the flexible joints on the dynamics.
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Let's consider the following values for the axial stiffness:
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We also consider here a nano-hexapod with the identified optimal actuator stiffness (\(k = 10^5\,[N/m]\)).
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\subsection{Direct Velocity Feedback}
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The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure \ref{fig:flex_joints_trans_study_dvf}.
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It is shown that the axial stiffness of the flexible joints does have a huge impact on the dynamics.
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If the axial stiffness of the flexible joints is \(K_a > 10^7\,[N/m]\) (here \(100\) times higher than the actuator stiffness), then the change of dynamics stays reasonably small.
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This is more clear by looking at the root locus (Figures \ref{fig:flex_joints_trans_study_dvf_root_locus} and \ref{fig:flex_joints_trans_study_root_locus_unzoom}).
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It can be seen that very little active damping can be achieve for axial stiffnesses below \(10^7\,[N/m]\).
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/flex_joints_trans_study_dvf.png}
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\caption{\label{fig:flex_joints_trans_study_dvf}Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered axis Stiffnesses}
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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]{figs/flex_joints_trans_study_dvf_root_locus.png}
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\caption{\label{fig:flex_joints_trans_study_dvf_root_locus}Root Locus for all the considered axial Stiffnesses}
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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]{figs/flex_joints_trans_study_root_locus_unzoom.png}
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\caption{\label{fig:flex_joints_trans_study_root_locus_unzoom}Root Locus (unzoom) for all the considered axial Stiffnesses}
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\end{figure}
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\subsection{Primary Control}
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The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure \ref{fig:flex_joints_trans_study_primary_plant}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/flex_joints_trans_study_primary_plant.png}
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\caption{\label{fig:flex_joints_trans_study_primary_plant}Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered axial stiffnesses}
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\end{figure}
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\section{Conclusion}
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\begin{important}
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The axial stiffness of the flexible joints should be maximized.
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For the considered actuator stiffness \(k = 10^5\,[N/m]\), the axial stiffness of the flexible joints should ideally be above \(10^7\,[N/m]\).
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This is a reasonable stiffness value for such joints.
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We may interpolate the results and say that the axial joint stiffness should be 100 times higher than the actuator stiffness, but this should be confirmed with further analysis.
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\end{important}
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\chapter{Designed Flexible Joints}
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\section{Initialization}
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Let's initialize all the stages with default parameters.
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Let's consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
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\section{Direct Velocity Feedback}
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\section{Integral Force Feedback}
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\chapter{Conclusion}
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\label{sec:conclusion}
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\begin{important}
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In this study we considered the bending, torsional and axial stiffnesses of the flexible joints used for the nano-hexapod.
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The bending and torsional stiffnesses somehow adds a parasitic stiffness in parallel with the legs.
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It is not found to be much problematic for the considered control architecture (it is however, if Integral Force Feedback is to be used).
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As a consequence of the added stiffness, it could increase a little bit the required actuator force.
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The axial stiffness of the flexible joints can be very problematic for control.
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Small values of the axial stiffness are shown to limit the achievable damping with Direct Velocity Feedback.
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The axial stiffness should therefore be maximized and taken into account in the model of the nano-hexapod.
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For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties:
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\begin{itemize}
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\item Axial Stiffness: \(K_a > 10^7\,[N/m]\)
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\item Bending Stiffness: \(K_b < 50\,[Nm/rad]\)
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\item Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)
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\end{itemize}
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As there is generally a trade-off between bending stiffness and axial stiffness, it should be highlighted that the \textbf{axial} stiffness is the most important property of the flexible joints.
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\end{important}
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\printbibliography[heading=bibintoc,title={Bibliography}]
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\end{document}
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