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Thomas Dehaeze 2025-04-02 17:04:27 +02:00
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nass-geometry.org
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@ -1355,7 +1355,7 @@ view(105, 15);
exportFig('figs/detail_kinematics_mobility_translation_strut_orientation.pdf', 'width', 'full', 'height', 'full', 'simplify', true);
#+end_src
#+name: fig:detail_kinematics_stewart_mobility_examples
#+name: fig:detail_kinematics_stewart_mobility_translation_examples
#+caption: Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometry are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}).
#+attr_latex: :options [htbp]
#+begin_figure
@ -1487,7 +1487,7 @@ view(105, 15);
exportFig('figs/detail_kinematics_mobility_angle_strut_distance.pdf', 'width', 'full', 'height', 'full', 'simplify', true);
#+end_src
#+name: fig:detail_kinematics_stewart_mobility_examples
#+name: fig:detail_kinematics_stewart_mobility_rotation_examples
#+caption: Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometry are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further appart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}).
#+attr_latex: :options [htbp]
#+begin_figure
@ -1574,17 +1574,17 @@ The dynamical equations (both in the Cartesian frame and in the frame of the str
The dynamics depend both on the geometry (Jacobian matrix) and on the payload being placed on top of the platform.
# Section ref:ssec:nhexa_stewart_platform_dynamics (page pageref:ssec:nhexa_stewart_platform_dynamics).
Under very specific conditions, the equations of motion in the Cartesian frame, given by equation eqref:eq:nhexa_transfer_function_cart, can be decoupled.
Under very specific conditions, the equations of motion in the Cartesian frame, given by equation eqref:eq:detail_kinematics_transfer_function_cart, can be decoupled.
These conditions are studied in Section ref:ssec:detail_kinematics_cubic_dynamic.
\begin{equation}\label{eq:nhexa_transfer_function_cart}
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{J} )^{-1}
\end{equation}
In the frame of the struts, the equations of motion given by equation eqref:eq:nhexa_transfer_function_struts are well decoupled at low frequency.
In the frame of the struts, the equations of motion given by equation eqref:eq:detail_kinematics_transfer_function_struts are well decoupled at low frequency.
This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is decoupled and SISO control may be applied for each strut, independently of the payload being used.
\begin{equation}\label{eq:nhexa_transfer_function_struts}
\begin{equation}\label{eq:detail_kinematics_transfer_function_struts}
\frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-T} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1}
\end{equation}
@ -2228,9 +2228,19 @@ The analysis aims to identify whether the cubic configuration exhibits special p
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 static behavior of the platform depends on the stiffness matrix eqref:eq:detail_kinematics_transfer_function_cart_low_freq.
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1}
\end{equation}
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.
In this case, the "Cartesian" plant is decoupled at low frequency.
At high frequency, the behavior is governed by the mass matrix (evaluated at frame $\{B\}$) eqref:eq:detail_kinematics_transfer_function_high_freq.
\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1}
\end{equation}
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.
#+name: fig:detail_kinematics_cubic_payload
@ -2655,7 +2665,7 @@ G_non_cubic.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
**** 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:nhexa_transfer_function_struts, the $6 \times 6$ plant is decoupled at low frequency.
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.
At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
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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nass-geometry.pdf
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nass-geometry.tex
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@ -1,4 +1,4 @@
% Created 2025-04-02 Wed 16:52
% Created 2025-04-02 Wed 17:04
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -231,7 +231,7 @@ The amplification factor increases when the struts have a high angle with the di
\end{center}
\subcaption{\label{fig:detail_kinematics_mobility_translation_strut_orientation}Translational mobility}
\end{subfigure}
\caption{\label{fig:detail_kinematics_stewart_mobility_examples}Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometry are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}).}
\caption{\label{fig:detail_kinematics_stewart_mobility_translation_examples}Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometry are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}).}
\end{figure}
\paragraph{Mobility in rotation}
@ -264,7 +264,7 @@ Having struts further apart decreases the ``lever arm'' and therefore reduces th
\end{center}
\subcaption{\label{fig:detail_kinematics_mobility_angle_strut_distance}Rotational mobility}
\end{subfigure}
\caption{\label{fig:detail_kinematics_stewart_mobility_examples}Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometry are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further appart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}).}
\caption{\label{fig:detail_kinematics_stewart_mobility_rotation_examples}Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometry are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further appart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}).}
\end{figure}
\paragraph{Combined translations and rotations}
@ -325,17 +325,17 @@ This is discussed in Section \ref{ssec:detail_kinematics_cubic_static}.
The dynamical equations (both in the Cartesian frame and in the frame of the struts) for the Stewart platform were derived during the conceptual phase with simplifying assumptions (massless struts and perfect joints).
The dynamics depend both on the geometry (Jacobian matrix) and on the payload being placed on top of the platform.
Under very specific conditions, the equations of motion in the Cartesian frame, given by equation \eqref{eq:nhexa_transfer_function_cart}, can be decoupled.
Under very specific conditions, the equations of motion in the Cartesian frame, given by equation \eqref{eq:detail_kinematics_transfer_function_cart}, can be decoupled.
These conditions are studied in Section \ref{ssec:detail_kinematics_cubic_dynamic}.
\begin{equation}\label{eq:nhexa_transfer_function_cart}
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{J} )^{-1}
\end{equation}
In the frame of the struts, the equations of motion given by equation \eqref{eq:nhexa_transfer_function_struts} are well decoupled at low frequency.
In the frame of the struts, the equations of motion given by equation \eqref{eq:detail_kinematics_transfer_function_struts} are well decoupled at low frequency.
This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is decoupled and SISO control may be applied for each strut, independently of the payload being used.
\begin{equation}\label{eq:nhexa_transfer_function_struts}
\begin{equation}\label{eq:detail_kinematics_transfer_function_struts}
\frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-T} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1}
\end{equation}
@ -545,9 +545,19 @@ The analysis aims to identify whether the cubic configuration exhibits special p
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 static behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}.
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1}
\end{equation}
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.
In this case, the ``Cartesian'' plant is decoupled at low frequency.
At high frequency, the behavior is governed by the mass matrix (evaluated at frame \(\{B\}\)) \eqref{eq:detail_kinematics_transfer_function_high_freq}.
\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1}
\end{equation}
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.
\begin{figure}[htbp]
@ -637,7 +647,7 @@ The second uses a non-cubic Stewart platform shown in Figure \ref{fig:detail_kin
\paragraph{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:nhexa_transfer_function_struts}, the \(6 \times 6\) plant is decoupled at low frequency.
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.
At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
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}).