470 lines
32 KiB
TeX
470 lines
32 KiB
TeX
% Created 2025-11-26 Wed 09:39
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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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\input{preamble_extra.tex}
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\bibliography{dehaeze26_cubic_architecture.bib}
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\author{Dehaeze Thomas}
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\date{\today}
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\title{Decoupling Properties of the Cubic Architecture}
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\hypersetup{
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pdfauthor={Dehaeze Thomas},
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pdftitle={Decoupling Properties of the Cubic Architecture},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 30.2 (Org mode 9.7.34)},
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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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The Cubic configuration for the Stewart platform was first proposed by Dr. Gough in a comment to the original paper by Dr. Stewart \cite{stewart65_platf_with_six_degrees_freed}.
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This configuration is characterized by active struts arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure \ref{fig:detail_kinematics_cubic_architecture_example}.
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Typically, the struts have similar length to the cube's edges, as illustrated in Figure \ref{fig:detail_kinematics_cubic_architecture_example}.
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Practical implementations of such configurations can be observed in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_uqp}.
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It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure \ref{fig:detail_kinematics_cubic_architecture_example_small}), as exemplified in Figure \ref{fig:detail_kinematics_ulb_pz}.
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\begin{figure}[htbp]
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_architecture_example}Classical Cubic architecture}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example_small.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}Alternative configuration}
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\end{subfigure}
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\caption{\label{fig:detail_kinematics_cubic_architecture_examples}Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges's length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}).}
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\end{figure}
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Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption \cite{geng94_six_degree_of_freed_activ,preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm}: simplified kinematics relationships and dynamical analysis \cite{geng94_six_degree_of_freed_activ}; uniform stiffness in all directions \cite{hanieh03_activ_stewar}; uniform mobility \cite[, chapt.8.5.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}; and minimization of the cross coupling between actuators and sensors in different struts \cite{preumont07_six_axis_singl_stage_activ}.
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This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control \cite{geng94_six_degree_of_freed_activ,thayer02_six_axis_vibrat_isolat_system}.
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These properties are examined in this section to assess their relevance for the nano-hexapod.
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The mobility and stiffness properties of the cubic configuration are analyzed in Section \ref{ssec:detail_kinematics_cubic_static}.
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Dynamical decoupling is investigated in Section \ref{ssec:detail_kinematics_cubic_dynamic}, while decentralized control, crucial for the NASS, is examined in Section \ref{ssec:detail_kinematics_decentralized_control}.
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Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section \ref{ssec:detail_kinematics_cubic_design}.
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The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod.
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\chapter{Static Properties}
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\label{ssec:detail_kinematics_cubic_static}
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\section{Stiffness matrix for the Cubic architecture}
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Consider the cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}.
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The unit vectors corresponding to the edges of the cube are described by equation \eqref{eq:detail_kinematics_cubic_s}.
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\begin{equation}\label{eq:detail_kinematics_cubic_s}
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\hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
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\hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
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\hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
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\hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
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\hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
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\hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
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\end{equation}
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\begin{figure}[htbp]
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_cubic_schematic_full.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_schematic_full}Full cube}
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\end{subfigure}
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_cubic_schematic.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_schematic}Cube's portion}
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\end{subfigure}
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\caption{\label{fig:detail_kinematics_cubic_schematic_cases}Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic})}
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\end{figure}
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Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center, are shown in equation \eqref{eq:detail_kinematics_cubic_vertices}.
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\begin{equation}\label{eq:detail_kinematics_cubic_vertices}
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\tilde{\bm{b}}_1 = \tilde{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad
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\tilde{\bm{b}}_3 = \tilde{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad
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\tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix}
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\end{equation}
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In the case where top joints are positioned at the cube's vertices, a diagonal stiffness matrix is obtained as shown in equation \eqref{eq:detail_kinematics_cubic_stiffness}.
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Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size \(H_c\).
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\begin{equation}\label{eq:detail_kinematics_cubic_stiffness}
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\bm{K}_{\{B\} = \{C\}} = k \begin{bmatrix}
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2 & 0 & 0 & 0 & 0 & 0 \\
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0 & 2 & 0 & 0 & 0 & 0 \\
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0 & 0 & 2 & 0 & 0 & 0 \\
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0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 & 0 \\
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0 & 0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 \\
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0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
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\end{bmatrix}
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\end{equation}
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However, typically, the top joints are not placed at the cube's vertices but at positions along the cube's edges (Figure \ref{fig:detail_kinematics_cubic_schematic}).
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In that case, the location of the top joints can be expressed by equation \eqref{eq:detail_kinematics_cubic_edges}, yet the computed stiffness matrix remains identical to Equation \eqref{eq:detail_kinematics_cubic_stiffness}.
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\begin{equation}\label{eq:detail_kinematics_cubic_edges}
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\bm{b}_i = \tilde{\bm{b}}_i + \alpha \hat{\bm{s}}_i
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\end{equation}
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The stiffness matrix is therefore diagonal when the considered \(\{B\}\) frame is located at the center of the cube (shown by frame \(\{C\}\)).
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This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center).
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This specific location where the stiffness matrix is diagonal is referred to as the ``Center of Stiffness'' (analogous to the ``Center of Mass'' where the mass matrix is diagonal).
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\section{Effect of having frame \(\{B\}\) off-centered}
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When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix.
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Considering a vertical shift as shown in Figure \ref{fig:detail_kinematics_cubic_schematic}, the stiffness matrix transforms into that shown in Equation \eqref{eq:detail_kinematics_cubic_stiffness_off_centered}.
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Off-diagonal elements increase proportionally with the height difference between the cube's center and the considered \(\{B\}\) frame.
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\begin{equation}\label{eq:detail_kinematics_cubic_stiffness_off_centered}
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\bm{K}_{\{B\} \neq \{C\}} = k \begin{bmatrix}
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2 & 0 & 0 & 0 & -2 H & 0 \\
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0 & 2 & 0 & 2 H & 0 & 0 \\
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0 & 0 & 2 & 0 & 0 & 0 \\
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0 & 2 H & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 & 0 \\
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-2 H & 0 & 0 & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 \\
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0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
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\end{bmatrix}
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\end{equation}
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This stiffness matrix structure is characteristic of Stewart platforms exhibiting symmetry, and is not an exclusive property of cubic architectures.
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Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center.
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This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform.
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It should be noted that for the stiffness matrix to be diagonal, the cube's center doesn't need to coincide with the geometric center of the Stewart platform.
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This observation leads to the interesting alternative architectures presented in Section \ref{ssec:detail_kinematics_cubic_design}.
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\section{Uniform Mobility}
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The translational mobility of the Stewart platform with constant orientation was analyzed.
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Considering limited actuator stroke (elongation of each strut), the maximum achievable positions in XYZ space were estimated.
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The resulting mobility in X, Y, and Z directions for the cubic architecture is illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_translations}.
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The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke.
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These findings suggest that the mobility pattern is more subtle than sometimes described in the literature \cite{mcinroy00_desig_contr_flexur_joint_hexap}, exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
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This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure \ref{fig:detail_kinematics_mobility_trans}.
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The rotational mobility, illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_rotations}, exhibits greater achievable angular stroke in the \(R_x\) and \(R_y\) directions compared to the \(R_z\) direction.
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Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
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\begin{figure}[htbp]
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_mobility_translations.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_mobility_translations}Mobility in translation}
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\end{subfigure}
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_mobility_rotations.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_mobility_rotations}Mobility in rotation}
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\end{subfigure}
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\caption{\label{fig:detail_kinematics_cubic_mobility}Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations})}
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\end{figure}
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\chapter{Dynamical Decoupling}
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\label{ssec:detail_kinematics_cubic_dynamic}
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This section examines the dynamics of the cubic architecture in the Cartesian frame which corresponds to the transfer function from forces and torques \(\bm{\mathcal{F}}\) to translations and rotations \(\bm{\mathcal{X}}\) of the top platform.
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When relative motion sensors are integrated in each strut (measuring \(\bm{\mathcal{L}}\)), the pose \(\bm{\mathcal{X}}\) is computed using the Jacobian matrix as shown in Figure \ref{fig:detail_kinematics_centralized_control}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png}
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\caption{\label{fig:detail_kinematics_centralized_control}Typical control architecture in the cartesian frame}
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\end{figure}
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\section{Low frequency and High frequency coupling}
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As derived during the conceptual design phase, the dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) is described by Equation \eqref{eq:detail_kinematics_transfer_function_cart}.
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At low frequency, the behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}.
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\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq}
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\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1}
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\end{equation}
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In Section \ref{ssec:detail_kinematics_cubic_static}, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame \(\{B\}\) is positioned at the cube's center.
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In this case, the ``Cartesian'' plant is decoupled at low frequency.
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At high frequency, the behavior is governed by the mass matrix (evaluated at frame \(\{B\}\)) \eqref{eq:detail_kinematics_transfer_function_high_freq}.
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\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq}
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\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1}
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\end{equation}
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To achieve a diagonal mass matrix, the center of mass of the mobile components must coincide with the \(\{B\}\) frame, and the principal axes of inertia must align with the axes of the \(\{B\}\) frame.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_cubic_payload.png}
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\caption{\label{fig:detail_kinematics_cubic_payload}Cubic stewart platform with top cylindrical payload}
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\end{figure}
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To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure \ref{fig:detail_kinematics_cubic_payload}).
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Transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) were computed for two specific locations of the \(\{B\}\) frames.
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When the \(\{B\}\) frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com}).
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Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_cok}).
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\begin{figure}[htbp]
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_com.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass}
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\end{subfigure}
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_cok.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center}
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\end{subfigure}
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\caption{\label{fig:detail_kinematics_cubic_cart_coupling}Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the \(\{B\}\) frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}).}
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\end{figure}
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\section{Payload's CoM at the cube's center}
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An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components \cite{li01_simul_fault_vibrat_isolat_point}.
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This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure \ref{fig:detail_kinematics_cubic_centered_payload}).
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This approach was physically implemented in several studies \cite{mcinroy99_dynam,jafari03_orthog_gough_stewar_platf_microm}, as shown in Figure \ref{fig:detail_kinematics_uw_gsp}.
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The resulting dynamics are indeed well-decoupled (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com_cok}), taking advantage from diagonal stiffness and mass matrices.
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The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform.
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If a design similar to Figure \ref{fig:detail_kinematics_cubic_centered_payload} were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation.
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\begin{figure}[htbp]
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_centered_payload.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_centered_payload}Payload at the cube's center}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_com_cok.png}
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\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com_cok}Fully decoupled cartesian plant}
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\end{subfigure}
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\caption{\label{fig:detail_kinematics_cubic_com_cok}Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok})}
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\end{figure}
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\section{Conclusion}
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The analysis of dynamical properties of the cubic architecture yields several important conclusions.
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Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames \(\{A\}\) and \(\{B\}\) are positioned at the cube's center.
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Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements.
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Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame \(\{B\}\).
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While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical.
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\chapter{Decentralized Control}
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\label{ssec:detail_kinematics_decentralized_control}
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The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control.
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Two sensor types integrated in the struts are considered: displacement sensors and force sensors.
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The control architecture is illustrated in Figure \ref{fig:detail_kinematics_decentralized_control}, where \(\bm{K}_{\mathcal{L}}\) represents a diagonal transfer function matrix.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/detail_kinematics_decentralized_control.png}
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\caption{\label{fig:detail_kinematics_decentralized_control}Decentralized control in the frame of the struts.}
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\end{figure}
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The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms.
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The first employs a cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_payload}.
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The second uses a non-cubic Stewart platform shown in Figure \ref{fig:detail_kinematics_non_cubic_payload}, featuring identical payload and strut dynamics but with struts oriented more vertically to differentiate it from the cubic architecture.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_non_cubic_payload.png}
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\caption{\label{fig:detail_kinematics_non_cubic_payload}Stewart platform with non-cubic architecture}
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\end{figure}
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\section{Relative Displacement Sensors}
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The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure \ref{fig:detail_kinematics_decentralized_dL}.
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As anticipated from the equations of motion from \(\bm{f}\) to \(\bm{\mathcal{L}}\) \eqref{eq:detail_kinematics_transfer_function_struts}, the \(6 \times 6\) plant is decoupled at low frequency.
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At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
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|
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No significant advantage is evident for the cubic architecture (Figure \ref{fig:detail_kinematics_cubic_decentralized_dL}) compared to the non-cubic architecture (Figure \ref{fig:detail_kinematics_non_cubic_decentralized_dL}).
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The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes.
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\begin{figure}[htbp]
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|
\begin{subfigure}{0.48\textwidth}
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|
\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_non_cubic_decentralized_dL.png}
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|
\end{center}
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\subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_dL}Non cubic architecture}
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\end{subfigure}
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\begin{subfigure}{0.48\textwidth}
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|
\begin{center}
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|
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_decentralized_dL.png}
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|
\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_decentralized_dL}Cubic architecture}
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\end{subfigure}
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\caption{\label{fig:detail_kinematics_decentralized_dL}Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL})}
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\end{figure}
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\section{Force Sensors}
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|
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Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
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The results are presented in Figure \ref{fig:detail_kinematics_decentralized_fn}.
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The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
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|
|
|
\begin{figure}[htbp]
|
|
\begin{subfigure}{0.48\textwidth}
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|
\begin{center}
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|
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_non_cubic_decentralized_fn.png}
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|
\end{center}
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\subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_fn}Non cubic architecture}
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\end{subfigure}
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|
\begin{subfigure}{0.48\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_decentralized_fn.png}
|
|
\end{center}
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\subcaption{\label{fig:detail_kinematics_cubic_decentralized_fn}Cubic architecture}
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\end{subfigure}
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|
\caption{\label{fig:detail_kinematics_decentralized_fn}Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn})}
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\end{figure}
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\section{Conclusion}
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|
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The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature.
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Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement for decentralized control is less obvious than often reported in the literature.
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\chapter{Cubic architecture with Cube's center above the top platform}
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\label{ssec:detail_kinematics_cubic_design}
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As demonstrated in Section \ref{ssec:detail_kinematics_cubic_dynamic}, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices.
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As shown in Section \ref{ssec:detail_kinematics_cubic_static}, the stiffness matrix is diagonal when the considered \(\{B\}\) frame is located at the cube's center.
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However, the \(\{B\}\) frame is typically positioned above the top platform where forces are applied and displacements are measured.
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|
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This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration.
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|
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|
Three key parameters define the geometry of the cubic Stewart platform: \(H\), the height of the Stewart platform (distance from fixed base to mobile platform); \(H_c\), the height of the cube, as shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}; and \(H_{CoM}\), the height of the center of mass relative to the mobile platform (coincident with the cube's center).
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|
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Depending on the cube's size \(H_c\) in relation to \(H\) and \(H_{CoM}\), different designs emerge.
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In the following examples, \(H = 100\,mm\) and \(H_{CoM} = 20\,mm\).
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\section{Small cube}
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|
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When the cube size \(H_c\) is smaller than twice the height of the CoM \(H_{CoM}\) \eqref{eq:detail_kinematics_cube_small}, the resulting design is shown in Figure \ref{fig:detail_kinematics_cubic_above_small}.
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\begin{equation}\label{eq:detail_kinematics_cube_small}
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H_c < 2 H_{CoM}
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\end{equation}
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|
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This configuration is similar to that described in \cite{furutani04_nanom_cuttin_machin_using_stewar}, although they do not explicitly identify it as a cubic configuration.
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Adjacent struts are parallel to each other, differing from the typical architecture where parallel struts are positioned opposite to each other.
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|
|
|
This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness.
|
|
|
|
\begin{figure}[htbp]
|
|
\begin{subfigure}{0.36\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_iso.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_small_iso}Isometric view}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.30\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_side.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_small_side}Side view}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.30\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_top.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_small_top}Top view}
|
|
\end{subfigure}
|
|
\caption{\label{fig:detail_kinematics_cubic_above_small}Cubic architecture with cube's center above the top platform. A cube height of 40mm is used.}
|
|
\end{figure}
|
|
\section{Medium sized cube}
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|
|
|
Increasing the cube's size such that \eqref{eq:detail_kinematics_cube_medium} is verified produces an architecture with intersecting struts (Figure \ref{fig:detail_kinematics_cubic_above_medium}).
|
|
|
|
\begin{equation}\label{eq:detail_kinematics_cube_medium}
|
|
2 H_{CoM} < H_c < 2 (H_{CoM} + H)
|
|
\end{equation}
|
|
|
|
This configuration resembles the design proposed in \cite{yang19_dynam_model_decoup_contr_flexib} (Figure \ref{fig:detail_kinematics_yang19}), although their design is not strictly cubic.
|
|
|
|
\begin{figure}[htbp]
|
|
\begin{subfigure}{0.36\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_iso.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_medium_iso}Isometric view}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.30\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_side.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_medium_side}Side view}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.30\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_top.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_medium_top}Top view}
|
|
\end{subfigure}
|
|
\caption{\label{fig:detail_kinematics_cubic_above_medium}Cubic architecture with cube's center above the top platform. A cube height of 140mm is used.}
|
|
\end{figure}
|
|
\section{Large cube}
|
|
|
|
When the cube's height exceeds twice the sum of the platform height and CoM height \eqref{eq:detail_kinematics_cube_large}, the architecture shown in Figure \ref{fig:detail_kinematics_cubic_above_large} is obtained.
|
|
|
|
\begin{equation}\label{eq:detail_kinematics_cube_large}
|
|
2 (H_{CoM} + H) < H_c
|
|
\end{equation}
|
|
|
|
\begin{figure}[htbp]
|
|
\begin{subfigure}{0.36\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_iso.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_large_iso}Isometric view}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.30\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_side.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_large_side}Side view}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.30\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_top.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:detail_kinematics_cubic_above_large_top}Top view}
|
|
\end{subfigure}
|
|
\caption{\label{fig:detail_kinematics_cubic_above_large}Cubic architecture with cube's center above the top platform. A cube height of 240mm is used.}
|
|
\end{figure}
|
|
\section{Platform size}
|
|
|
|
For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom joints \(\bm{a}_i\)) and are positioned on a circle with radius \(R_{b_i}\) (resp. \(R_{a_i}\)) described by Equation \eqref{eq:detail_kinematics_cube_joints}.
|
|
|
|
\begin{subequations}\label{eq:detail_kinematics_cube_joints}
|
|
\begin{align}
|
|
R_{b_i} &= \sqrt{\frac{3}{2} H_c^2 + 2 H_{CoM}^2} \label{eq:detail_kinematics_cube_top_joints} \\
|
|
R_{a_i} &= \sqrt{\frac{3}{2} H_c^2 + 2 (H_{CoM} + H)^2} \label{eq:detail_kinematics_cube_bot_joints}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
Since the rotational stiffness for the cubic architecture scales with the square of the cube's height \eqref{eq:detail_kinematics_cubic_stiffness}, the cube's size can be determined based on rotational stiffness requirements.
|
|
Subsequently, using Equation \eqref{eq:detail_kinematics_cube_joints}, the dimensions of the top and bottom platforms can be calculated.
|
|
\chapter*{Conclusion}
|
|
The analysis of the cubic architecture for Stewart platforms yielded several important findings.
|
|
While the cubic configuration provides uniform stiffness in the XYZ directions, it stiffness property becomes particularly advantageous when forces and torques are applied at the cube's center.
|
|
Under these conditions, the stiffness matrix becomes diagonal, resulting in a decoupled Cartesian plant at low frequencies.
|
|
|
|
Regarding mobility, the translational capabilities of the cubic configuration exhibit uniformity along the directions of the orthogonal struts, rather than complete uniformity in the Cartesian space.
|
|
This understanding refines the characterization of cubic architecture mobility commonly presented in literature.
|
|
|
|
The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected.
|
|
While cubic architectures are frequently associated with reduced coupling between actuators and sensors, this study showed that these benefits may be more subtle or context-dependent than commonly described.
|
|
Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations, in the frame of the struts, appeared similar.
|
|
|
|
Fully decoupled dynamics in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center.
|
|
However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications.
|
|
|
|
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform.
|
|
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
|
|
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
|
|
\printbibliography[heading=bibintoc,title={Bibliography}]
|
|
\end{document}
|