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paper/dehaeze26_cubic_architecture.tex
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% Created 2025-11-26 Wed 09:39
% Created 2025-11-27 Thu 17:52
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
\documentclass[balance,nocopyright,nolists]{asmejour}
\input{preamble.tex}
\input{preamble_extra.tex}
\bibliography{dehaeze26_cubic_architecture.bib}
\usepackage[square,numbers]{natbib}
\JourName{Journal of Mechanical Design}
\author{Dehaeze Thomas}
\date{\today}
\title{Decoupling Properties of the Cubic Architecture}
\hypersetup{
pdfauthor={Dehaeze Thomas},
pdftitle={Decoupling Properties of the Cubic Architecture},
pdftitle={},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 30.2 (Org mode 9.7.34)},
pdflang={English}}
\usepackage{biblatex}
\begin{document}
\maketitle
\tableofcontents
\SetAuthorBlock{Thomas Dehaeze\CorrespondingAuthor}{%
ESRF, The European Synchrotron\\
Grenoble, France\\
email: thomas.dehaeze@esrf.fr
}
\SetAuthorBlock{Christophe Collette}{%
Precision Mechatronics Laboratory\\
University of Li\`{e}ge, Belgium\\
email: christophe.collette@uliege.be
}
%%% Change to your paper title. Can insert line breaks if you wish (otherwise breaks are selected automatically).
\title{Decoupling Properties of the Cubic Architecture}
\begin{abstract}
{\it This is the abstract.
This article illustrates preparation of ASME paper using
Please use this template to test how your figures will look on the printed journal page of the Journal of Mechanical Design. The Journal will no longer publish papers that contain errors in figure resolution. These usually consist of unreadable or fuzzy text, and pixilation or rasterization of lines. This template identifies the specifications used by JMD some of which may not be easily duplicated; for example, ASME actually uses Helvetica Condensed Bold, but this is not generally available so for the purpose of this exercise Helvetica is adequate. However, reproduction of the journal page is not the goal, instead this exercise is to verify the quality of your figures. Notice that this abstract is to be set in 9pt Times Italic, single spaced and right justified.
}
\end{abstract}
\date{Version \versionno, \today}%% You can modify this information as desired.
%% Putting \date{} will suppress any date.
%% If this command is omitted, date defaults to \today
%% This command must come somewhere before \maketitle
\maketitle %% This command creates the author/title/abstract block. Essential!
\section{Introduction}
\clearpage
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}.
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}.
@@ -31,15 +56,15 @@ Practical implementations of such configurations can be observed in Figures \ref
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}.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example.png}
\includegraphics[scale=1,scale=0.7]{figs/detail_kinematics_cubic_architecture_example.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_architecture_example}Classical Cubic architecture}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example_small.png}
\includegraphics[scale=1,scale=0.7]{figs/detail_kinematics_cubic_architecture_example_small.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}Alternative configuration}
\end{subfigure}
@@ -55,32 +80,34 @@ The mobility and stiffness properties of the cubic configuration are analyzed in
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}.
Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section \ref{ssec:detail_kinematics_cubic_design}.
The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod.
\chapter{Static Properties}
\section{Static Properties}
\label{ssec:detail_kinematics_cubic_static}
\section{Stiffness matrix for the Cubic architecture}
\subsection{Stiffness matrix for the Cubic architecture}
Consider the cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}.
The unit vectors corresponding to the edges of the cube are described by equation \eqref{eq:detail_kinematics_cubic_s}.
\begin{equation}\label{eq:detail_kinematics_cubic_s}
\hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
\begin{align*}
\hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \\
\hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
\end{align*}
\end{equation}
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_cubic_schematic_full.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_schematic_full.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_schematic_full}Full cube}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_cubic_schematic.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_schematic.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_schematic}Cube's portion}
\end{subfigure}
@@ -90,16 +117,18 @@ The unit vectors corresponding to the edges of the cube are described by equatio
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}.
\begin{equation}\label{eq:detail_kinematics_cubic_vertices}
\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
\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
\tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix}
\begin{align*}
\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
\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}, \\
\tilde{\bm{b}}_5 &= \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix}
\end{align*}
\end{equation}
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}.
Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size \(H_c\).
\begin{equation}\label{eq:detail_kinematics_cubic_stiffness}
\bm{K}_{\{B\} = \{C\}} = k \begin{bmatrix}
\bm{K} = k \begin{bmatrix}
2 & 0 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 & 0 & 0 \\
0 & 0 & 2 & 0 & 0 & 0 \\
@@ -120,7 +149,7 @@ In that case, the location of the top joints can be expressed by equation \eqref
The stiffness matrix is therefore diagonal when the considered \(\{B\}\) frame is located at the center of the cube (shown by frame \(\{C\}\)).
This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center).
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).
\section{Effect of having frame \(\{B\}\) off-centered}
\subsection{Effect of having frame \(\{B\}\) off-centered}
When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix.
@@ -128,14 +157,15 @@ Considering a vertical shift as shown in Figure \ref{fig:detail_kinematics_cubic
Off-diagonal elements increase proportionally with the height difference between the cube's center and the considered \(\{B\}\) frame.
\begin{equation}\label{eq:detail_kinematics_cubic_stiffness_off_centered}
\bm{K}_{\{B\} \neq \{C\}} = k \begin{bmatrix}
\bm{K} = k \\
{\footnotesize \begin{bmatrix}
2 & 0 & 0 & 0 & -2 H & 0 \\
0 & 2 & 0 & 2 H & 0 & 0 \\
0 & 0 & 2 & 0 & 0 & 0 \\
0 & 2 H & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 & 0 \\
-2 H & 0 & 0 & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 \\
0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
\end{bmatrix}
\end{bmatrix}}
\end{equation}
This stiffness matrix structure is characteristic of Stewart platforms exhibiting symmetry, and is not an exclusive property of cubic architectures.
@@ -144,7 +174,7 @@ This poses a practical limitation, as in most applications, the relevant frame (
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.
This observation leads to the interesting alternative architectures presented in Section \ref{ssec:detail_kinematics_cubic_design}.
\section{Uniform Mobility}
\subsection{Uniform Mobility}
The translational mobility of the Stewart platform with constant orientation was analyzed.
Considering limited actuator stroke (elongation of each strut), the maximum achievable positions in XYZ space were estimated.
@@ -158,21 +188,21 @@ The rotational mobility, illustrated in Figure \ref{fig:detail_kinematics_cubic_
Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_mobility_translations.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_mobility_translations.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_mobility_translations}Mobility in translation}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_mobility_rotations.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_mobility_rotations.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_mobility_rotations}Mobility in rotation}
\end{subfigure}
\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})}
\end{figure}
\chapter{Dynamical Decoupling}
\section{Dynamical Decoupling}
\label{ssec:detail_kinematics_cubic_dynamic}
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.
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}.
@@ -182,7 +212,7 @@ When relative motion sensors are integrated in each strut (measuring \(\bm{\math
\includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png}
\caption{\label{fig:detail_kinematics_centralized_control}Typical control architecture in the cartesian frame}
\end{figure}
\section{Low frequency and High frequency coupling}
\subsection{Low frequency and High frequency coupling}
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}.
At low frequency, the behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}.
@@ -203,7 +233,7 @@ To achieve a diagonal mass matrix, the center of mass of the mobile components m
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_cubic_payload.png}
\includegraphics[scale=1,width=0.7\linewidth]{figs/detail_kinematics_cubic_payload.png}
\caption{\label{fig:detail_kinematics_cubic_payload}Cubic stewart platform with top cylindrical payload}
\end{figure}
@@ -213,52 +243,52 @@ When the \(\{B\}\) frame was positioned at the center of mass, coupling at low f
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}).
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_com.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_cart_coupling_com.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_cok.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_cart_coupling_cok.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center}
\end{subfigure}
\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}).}
\end{figure}
\section{Payload's CoM at the cube's center}
\subsection{Payload's CoM at the cube's center}
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}.
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}).
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}.
This approach was physically implemented in several studies \cite{mcinroy99_dynam,jafari03_orthog_gough_stewar_platf_microm}.
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.
The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform.
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.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_centered_payload.png}
\includegraphics[scale=1,width=0.7\linewidth]{figs/detail_kinematics_cubic_centered_payload.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_centered_payload}Payload at the cube's center}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_com_cok.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_cart_coupling_com_cok.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com_cok}Fully decoupled cartesian plant}
\end{subfigure}
\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})}
\end{figure}
\section{Conclusion}
\subsection{Conclusion}
The analysis of dynamical properties of the cubic architecture yields several important conclusions.
Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames \(\{A\}\) and \(\{B\}\) are positioned at the cube's center.
Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements.
Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame \(\{B\}\).
While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical.
\chapter{Decentralized Control}
\section{Decentralized Control}
\label{ssec:detail_kinematics_decentralized_control}
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.
Two sensor types integrated in the struts are considered: displacement sensors and force sensors.
@@ -276,10 +306,10 @@ The second uses a non-cubic Stewart platform shown in Figure \ref{fig:detail_kin
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_non_cubic_payload.png}
\includegraphics[scale=1,width=0.7\linewidth]{figs/detail_kinematics_non_cubic_payload.png}
\caption{\label{fig:detail_kinematics_non_cubic_payload}Stewart platform with non-cubic architecture}
\end{figure}
\section{Relative Displacement Sensors}
\subsection{Relative Displacement Sensors}
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}.
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.
@@ -289,46 +319,46 @@ No significant advantage is evident for the cubic architecture (Figure \ref{fig:
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.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_non_cubic_decentralized_dL.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_non_cubic_decentralized_dL.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_dL}Non cubic architecture}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_decentralized_dL.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_decentralized_dL.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_decentralized_dL}Cubic architecture}
\end{subfigure}
\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})}
\end{figure}
\section{Force Sensors}
\subsection{Force Sensors}
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
The results are presented in Figure \ref{fig:detail_kinematics_decentralized_fn}.
The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_non_cubic_decentralized_fn.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_non_cubic_decentralized_fn.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_fn}Non cubic architecture}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_decentralized_fn.png}
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_decentralized_fn.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_decentralized_fn}Cubic architecture}
\end{subfigure}
\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})}
\end{figure}
\section{Conclusion}
\subsection{Conclusion}
The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature.
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.
\chapter{Cubic architecture with Cube's center above the top platform}
\section{Cubic architecture with Cube's center above the top platform}
\label{ssec:detail_kinematics_cubic_design}
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.
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.
@@ -340,7 +370,7 @@ Three key parameters define the geometry of the cubic Stewart platform: \(H\), t
Depending on the cube's size \(H_c\) in relation to \(H\) and \(H_{CoM}\), different designs emerge.
In the following examples, \(H = 100\,mm\) and \(H_{CoM} = 20\,mm\).
\section{Small cube}
\subsection{Small cube}
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}.
@@ -354,27 +384,27 @@ Adjacent struts are parallel to each other, differing from the typical architect
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{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_iso.png}
\includegraphics[scale=1,scale=0.7]{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{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_side.png}
\includegraphics[scale=1,scale=0.7]{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{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_small_top.png}
\includegraphics[scale=1,scale=0.7]{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}
\subsection{Medium sized cube}
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}).
@@ -382,30 +412,30 @@ Increasing the cube's size such that \eqref{eq:detail_kinematics_cube_medium} is
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.
This configuration resembles the design proposed in \cite{yang19_dynam_model_decoup_contr_flexib}, although their design is not strictly cubic.
\begin{figure}[htbp]
\begin{subfigure}{0.36\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_iso.png}
\includegraphics[scale=1,scale=0.7]{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{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_side.png}
\includegraphics[scale=1,scale=0.7]{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{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_medium_top.png}
\includegraphics[scale=1,scale=0.7]{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}
\subsection{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.
@@ -414,27 +444,27 @@ When the cube's height exceeds twice the sum of the platform height and CoM heig
\end{equation}
\begin{figure}[htbp]
\begin{subfigure}{0.36\textwidth}
\begin{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_iso.png}
\includegraphics[scale=1,scale=0.7]{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{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_side.png}
\includegraphics[scale=1,scale=0.7]{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{subfigure}{\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_kinematics_cubic_above_large_top.png}
\includegraphics[scale=1,scale=0.7]{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}
\subsection{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}.
@@ -447,7 +477,7 @@ For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom jo
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}
\section*{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.
@@ -465,5 +495,17 @@ However, this arrangement presents practical challenges, as the cube's center is
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}]
\begin{acknowledgment}
ASME Technical Publications provided the format specifications for the Journal of Mechanical Design, though they are not easy to reproduce. It is their commitment to ensuring quality figures in every issue of JMD that motivates this effort to have authors review the presentation of their figures.
Thanks go to D. E. Knuth and L. Lamport for developing the wonderful word processing software packages \TeX\ and \LaTeX. We would like to thank Ken Sprott, Kirk van Katwyk, and Matt Campbell for fixing bugs in the ASME style file \verb+asme2ej.cls+, and Geoff Shiflett for creating
ASME bibliography stype file \verb+asmems4.bst+.
\end{acknowledgment}
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% article is asme2e.bib. The name for your database is up
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\end{document}