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Second draft before full writing

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Thomas Dehaeze 2025-04-01 19:15:19 +02:00
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% Created 2025-04-01 Tue 14:18
% Created 2025-04-01 Tue 17:49
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -27,7 +27,7 @@
\begin{itemize}
\item In the conceptual design phase, the geometry of the Stewart platform was chosen arbitrarily and not optimized
\item In the detail design phase, we want to see if the geometry can be optimized to improve the overall performances
\item Optimization criteria: mobility, stiffness, dynamical decoupling, more performance / bandwidth
\item Optimization criteria: mobility, stiffness, decoupling between the struts for decentralized control, dynamical decoupling in the cartesian frame
\end{itemize}
Outline:
@ -35,8 +35,8 @@ Outline:
\item Review of Stewart platform (Section \ref{sec:detail_kinematics_stewart_review})
Geometry, Actuators, Sensors, Joints
\item Effect of geometry on the Stewart platform characteristics (Section \ref{sec:detail_kinematics_geometry})
\item Cubic configuration: benefits? (Section \ref{sec:detail_kinematics_cubic})
\item Obtained geometry for the nano hexapod (Section \ref{sec:detail_kinematics_nano_hexapod})
\item Cubic configuration: special architecture that received many attention in the literature. We want to see the special properties of this architecture and if this can be applied for the nano hexapod (Section \ref{sec:detail_kinematics_cubic})
\item Presentation of the obtained geometry for the nano hexapod (Section \ref{sec:detail_kinematics_nano_hexapod})
\end{itemize}
\chapter{Review of Stewart platforms}
@ -60,14 +60,17 @@ Some Stewart platforms found in the literature are listed in Table \ref{tab:deta
\item All presented Stewart platforms are using flexible joints, as it is a prerequisites for nano-positioning capabilities.
\item Most of stewart platforms are using voice coil actuators or piezoelectric actuators.
The actuators used for the Stewart platform will be chosen in the next section.
\item Depending on the application, various sensors are integrated in the struts or on the plates.
\item Depending on the application, various sensors are integrated in the struts or on the plates such as force sensors, inertial sensors or relative displacement sensors.
The choice of sensor for the nano-hexapod will be described in the next section.
\item There are two categories of Stewart platform geometry:
\item Flexible joints can also have various implementations. This will be discussed in the next section.
\item There are two main categories of Stewart platform geometry:
\begin{itemize}
\item Cubic architecture (Figure \ref{fig:detail_kinematics_stewart_examples_cubic}).
Struts are positioned along 6 sides of a cubic (and are therefore orthogonal to each other).
Such specific architecture has some special properties that will be studied in Section \ref{sec:detail_kinematics_cubic}.
\item Non-cubic architecture (Figure \ref{fig:detail_kinematics_stewart_examples_non_cubic})
The orientation of the struts and position of the joints are chosen based on performances criteria.
Some of which are presented in Section \ref{sec:detail_kinematics_geometry}
\end{itemize}
\end{itemize}
@ -170,7 +173,7 @@ Conclusion:
\item Lot's have a ``cubic'' architecture that will be discussed in Section \ref{sec:detail_kinematics_cubic}
\item actuator types
\item various sensors
\item flexible joints (discussed in next chapter)
\item flexible joints
\end{itemize}
\item The effect of geometry on the properties of the Stewart platform is studied in section \ref{sec:detail_kinematics_geometry}
\item It is determined what is the optimal geometry for the NASS
@ -182,7 +185,7 @@ Conclusion:
\item As was shown during the conceptual phase, the geometry of the Stewart platform influences:
\begin{itemize}
\item the stiffness and compliance properties
\item the mobility
\item the mobility or workspace
\item the force authority
\item the dynamics of the manipulator
\end{itemize}
@ -194,7 +197,12 @@ The choice of frames (\(\{A\}\) and \(\{B\}\)), independently of the physical St
\section{Platform Mobility / Workspace}
The mobility of the Stewart platform (or any manipulator) is here defined as the range of motion that it can perform.
It corresponds to the set of possible pose (i.e. combined translation and rotation) of frame \{B\} with respect to frame \{A\}.
It should therefore be represented in a six dimensional space.
It is therefore a six dimensional property which is difficult to represent.
Depending on the applications, only the translation mobility (i.e. fixed orientation workspace) or the rotation mobility may be represented.
This is equivalent as to project the six dimensional value into a 3 dimensional space, easier to represent.
Mobility of parallel manipulators are inherently difficult to study as the translational and orientation workspace are coupled \cite{merlet02_still}.
Things are getting much more simpler when considering small motions as the Jacobian matrix can be considered constant and the equations are linear.
As was shown during the conceptual phase, for small displacements, the Jacobian matrix can be used to link the strut motion to the motion of frame B with respect to A through equation \eqref{eq:detail_kinematics_jacobian}.
\begin{equation}\label{eq:detail_kinematics_jacobian}
@ -220,34 +228,18 @@ More specifically:
\item the mobility in rotation depends on bi (position of top joints)
\end{itemize}
As will be shown in Section \ref{sec:detail_kinematics_cubic}, there are some geometry that gives same stroke in X, Y and Z directions.
As the mobility is of dimension six, it is difficult to represent.
Depending on the applications, only the translation mobility or the rotation mobility may be represented.
\begin{quote}
Although there is no human readable way to represent the complete workspace, some projections of the full workspace can be drawn.
\end{quote}
Difficulty of studying workspace of parallel manipulators.
\paragraph{Mobility in translation}
Here, for simplicity, only translations are first considered:
Here, for simplicity, only translations are first considered (i.e. fixed orientation of the Stewart platform):
\begin{itemize}
\item Let's consider a general Stewart platform geometry shown in Figure \ref{fig:detail_kinematics_mobility_trans_arch}.
\item In the general case: the translational mobility can be represented by a 3D shape with 12 faces (each actuator limits the stroke along its orientation in positive and negative directions).
The faces are therefore perpendicular to the strut direction.
The obtained mobility is shown in Figure \ref{fig:detail_kinematics_mobility_trans_result}.
The obtained mobility for the considered stewart platform geometry is shown in Figure \ref{fig:detail_kinematics_mobility_trans_result}.
In reality, the workspace boundaries are portion of spheres, but they are well approximated by flat surfaces for short stroke hexapods
\item Considering an actuator stroke of \(\pm d\), the mobile platform can be translated in any direction with a stroke of \(d\)
A circle with radius \(d\) can be contained in the general shape.
It will touch the shape along six lines defined by the strut axes.
The sphere with radius \(d\) is shown in Figure \ref{fig:detail_kinematics_mobility_trans_result}.
\item Therefore, for any (small stroke) Stewart platform with actuator stroke \(\pm d\), it is possible to move the top platform in any direction by at least a distance \(d\).
Note that no platform angular motion is here considered. When combining angular motion, the linear stroke decreases.
\item When considering some symmetry in the system (as typically the case), the shape becomes a Trigonal trapezohedron whose height and width depends on the orientation of the struts.
We only get 6 faces as usually the Stewart platform consists of 3 sets of 2 parallels struts.
\item In reality, portion of spheres, but well approximated by flat surfaces for short stroke hexapods.
This means that a sphere with radius \(d\) is contained in the general shape as illustrated in Figure \ref{fig:detail_kinematics_mobility_trans_result}.
The sphere will touch the shape along six lines defined by the strut axes.
\end{itemize}
\begin{figure}[htbp]
@ -268,11 +260,16 @@ We only get 6 faces as usually the Stewart platform consists of 3 sets of 2 para
To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared:
\begin{itemize}
\item Struts oriented horizontally (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}) => more stroke in horizontal direction
\item Struts oriented vertically (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}) => more stroke in vertical direction
\item Corresponding mobility shown in Figure \ref{fig:detail_kinematics_mobility_translation_strut_orientation}
\item Struts oriented horizontally (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}).
This leads to having more stroke in the horizontal direction and less stroke in the vertical direction (Figure \ref{fig:detail_kinematics_mobility_translation_strut_orientation}).
\item Struts oriented vertically (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}).
More stroke in vertical direction
\end{itemize}
It can be counter intuitive to have less stroke in the direction of the struts.
This is because the struts are forming a lever mechanism that amplifies the motion.
The amplification factor increases when the struts have an high angle with the direction and motion and is equal to one when it is aligned with the direction of motion.
\begin{figure}[htbp]
\begin{subfigure}{0.25\textwidth}
\begin{center}
@ -352,8 +349,8 @@ It will be done in Section \ref{sec:detail_kinematics_nano_hexapod} to estimate
Stiffness matrix:
\begin{itemize}
\item defines how the nano-hexapod deforms (frame \(\{B\}\) with respect to frame \(\{A\}\)) due to static forces/torques applied on \(\{B\}\).
\item Depends on the Jacobian matrix (i.e. the geometry) and the strut axial stiffness \eqref{eq:detail_kinematics_stiffness_matrix}
\item Contribution of joints stiffness is here not considered \cite{mcinroy00_desig_contr_flexur_joint_hexap,mcinroy02_model_desig_flexur_joint_stewar}
\item It depends on the Jacobian matrix (i.e. the geometry) and the strut axial stiffness \eqref{eq:detail_kinematics_stiffness_matrix}
\item The contribution of joints stiffness is here not considered \cite{mcinroy00_desig_contr_flexur_joint_hexap,mcinroy02_model_desig_flexur_joint_stewar}
\end{itemize}
\begin{equation}\label{eq:detail_kinematics_stiffness_matrix}
@ -361,7 +358,7 @@ Stiffness matrix:
\end{equation}
It is assumed that the stiffness of all strut is the same: \(\bm{\mathcal{K}} = k \cdot \mathbf{I}_6\).
Obtained stiffness matrix linearly depends on the strut stiffness \(k\) \eqref{eq:detail_kinematics_stiffness_matrix_simplified}.
Obtained stiffness matrix linearly depends on the strut stiffness \(k\), and is structured as shown in \eqref{eq:detail_kinematics_stiffness_matrix_simplified}.
\begin{equation}\label{eq:detail_kinematics_stiffness_matrix_simplified}
\bm{K} = k \bm{J}^T \bm{J} =
@ -376,13 +373,13 @@ Obtained stiffness matrix linearly depends on the strut stiffness \(k\) \eqref{e
\paragraph{Translation Stiffness}
XYZ stiffnesses:
As shown by \eqref{eq:detail_kinematics_stiffness_matrix_simplified}, the translation stiffnesses (the 3x3 top left terms of the stiffness matrix):
\begin{itemize}
\item Only depends on the orientation of the struts and not their location: \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T\)
\item Extreme case: all struts are vertical \(s_i = [0,\ 0,\ 1]\) => vertical stiffness of \(6 k\), but null stiffness in X and Y directions
\item If two struts along X, two struts along Y, and two struts along Z => \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3\)
Stiffness is well distributed along directions.
This corresponds to the cubic architecture.
This corresponds to the cubic architecture presented in Section \ref{sec:detail_kinematics_cubic}.
\end{itemize}
If struts more vertical (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}):
@ -397,26 +394,16 @@ Opposite conclusions if struts are not horizontal (Figure \ref{fig:detail_kinema
\paragraph{Rotational Stiffness}
Rotational stiffnesses:
\begin{itemize}
\item Same orientation but increased distances (bi) by a factor 2 => rotational stiffness increased by factor 4
Figure \ref{fig:detail_kinematics_stewart_mobility_close_struts}
Figure \ref{fig:detail_kinematics_stewart_mobility_space_struts}
\end{itemize}
The rotational stiffnesses depends both on the orientation of the struts and on the location of the top joints (with respect to the considered center of rotation, i.e. the location of frame B).
Struts further apart:
\begin{itemize}
\item no change to XYZ
\item increase in rotational stiffness (by the square of the distance)
\end{itemize}
Same orientation but increased distances (bi) by a factor 2 => rotational stiffness increased by factor 4.
Compact stewart platform of Figure \ref{fig:detail_kinematics_stewart_mobility_close_struts} as therefore less rotational stiffness than the Stewart platform of Figure \ref{fig:detail_kinematics_stewart_mobility_space_struts}.
\paragraph{Diagonal Stiffness Matrix}
Having the stiffness matrix \(\bm{K}\) diagonal can be beneficial for control purposes as it would make the plant in the cartesian frame decoupled at low frequency.
This depends on the geometry and on the chosen \{A\} frame.
For specific configurations, it is possible to have a diagonal K matrix.
This depends on the geometry and on the chosen \{B\} frame.
For specific geometry and chose of B frame, it is possible to have a diagonal K matrix.
This will be discussed in Section \ref{ssec:detail_kinematics_cubic_static}.
@ -426,7 +413,7 @@ This will be discussed in Section \ref{ssec:detail_kinematics_cubic_static}.
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 depends both on the geometry (Jacobian matrix) but also on the payload being placed on top of the platform.
Under very specific conditions, the equations of motion can be decoupled in the Cartesian space.
Under very specific conditions, the equations of motion in the Cartesian frame \eqref{eq:nhexa_transfer_function_cart} can be decoupled.
These are studied in Section \ref{ssec:detail_kinematics_cubic_dynamic}.
\begin{equation}\label{eq:nhexa_transfer_function_cart}
@ -435,21 +422,15 @@ These are studied in Section \ref{ssec:detail_kinematics_cubic_dynamic}.
\paragraph{In the frame of the Struts}
In the frame of the struts, the equations of motion are well decoupled at low frequency.
This is why most of Stewart platforms are controlled in the frame of the struts: bellow the resonance frequency, the system is decoupled and SISO control may be applied for each strut.
In the frame of the struts, the equations of motion \eqref{eq:nhexa_transfer_function_struts} are well decoupled at low frequency.
This is why most of Stewart platforms are controlled in the frame of the struts: bellow 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}
\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}
Coupling between sensors (force sensors, relative position sensor, inertial sensors) in different struts may also be important for decentralized control.
Can the geometry be optimized to have lower coupling between the struts?
This will be studied with the cubic architecture.
\paragraph{Dynamic Isotropy}
\cite{afzali-far16_vibrat_dynam_isotr_hexap_analy_studies}:
``\textbf{Dynamic isotropy}, leading to equal eigenfrequencies, is a powerful optimization measure.''
In section \ref{ssec:detail_kinematics_decentralized_control}, it will be study if the Stewart platform geometry can be optimized to have lower coupling between the struts.
\section*{Conclusion}
The effects of two changes in the manipulator's geometry, namely the position and orientation of the legs, are summarized in Table \ref{tab:detail_kinematics_geometry}.
@ -460,20 +441,16 @@ These trade-offs give some guidelines when choosing the Stewart platform geometr
\begin{table}[htbp]
\caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness, force authority and stroke}
\centering
\begin{tabularx}{\linewidth}{lXX}
\small
\begin{tabularx}{0.9\linewidth}{Xcc}
\toprule
& \textbf{legs pointing more vertically} & \textbf{legs further apart}\\
\textbf{Struts} & \textbf{Vertically Oriented} & \textbf{Increased separation}\\
\midrule
Vertical stiffness & \(\nearrow\) & \(=\)\\
Horizontal stiffness & \(\searrow\) & \(=\)\\
Vertical rotation stiffness & \(\searrow\) & \(\nearrow\)\\
Horizontal rotation stiffness & \(\nearrow\) & \(\nearrow\)\\
\midrule
Vertical force authority & \(\nearrow\) & \(=\)\\
Horizontal force authority & \(\searrow\) & \(=\)\\
Vertical torque authority & \(\searrow\) & \(\nearrow\)\\
Horizontal torque authority & \(\nearrow\) & \(\nearrow\)\\
\midrule
Vertical stroke & \(\searrow\) & \(=\)\\
Horizontal stroke & \(\nearrow\) & \(=\)\\
Vertical rotation stroke & \(\nearrow\) & \(\searrow\)\\
@ -485,20 +462,13 @@ Horizontal rotation stroke & \(\searrow\) & \(\searrow\)\\
\chapter{The Cubic Architecture}
\label{sec:detail_kinematics_cubic}
The Cubic configuration for the Stewart platform was first proposed in \cite{geng94_six_degree_of_freed_activ}.
This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure \ref{fig:detail_kinematics_cubic_architecture_examples}.
This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube Figure \ref{fig:detail_kinematics_cubic_architecture_examples}.
Typically, the struts have similar size than the cube's edge, as shown in Figure \ref{fig:detail_kinematics_cubic_architecture_example}.
Practical implementations of such configuration are shown in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_uqp}.
Cubic configuration:
\begin{itemize}
\item The struts are corresponding to 6 of the 8 edges of a cube.
\item This way, all struts are perpendicular to each other (except sets of two that are parallel).
\end{itemize}
Struts with similar size than the cube's edge (Figure \ref{fig:detail_kinematics_cubic_architecture_example}).
Similar to Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_uqp}.
Struts smaller than the cube's edge (Figure \ref{fig:detail_kinematics_cubic_architecture_example_small}).
Similar to the Stewart platform of Figure \ref{fig:detail_kinematics_ulb_pz}.
It is also possible to have the struts length smaller than the cube's edge (Figure \ref{fig:detail_kinematics_cubic_architecture_example_small}).
An example of such Stewart platform is shown in Figure \ref{fig:detail_kinematics_ulb_pz}.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@ -517,36 +487,31 @@ Similar to the Stewart platform of Figure \ref{fig:detail_kinematics_ulb_pz}.
\end{figure}
The cubic configuration is attributed a number of properties that made this configuration widely used (\cite{preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm}).
From \cite{geng94_six_degree_of_freed_activ}:
\begin{enumerate}
\item Uniformity in control capability in all directions
\item Uniformity in stiffness in all directions
\item Minimum cross coupling force effect among actuators
\item Facilitate collocated sensor-actuator control system design
\item Simple kinematics relationships
\item Simple dynamic analysis
\item Simple mechanical design
\end{enumerate}
According to \cite{preumont07_six_axis_singl_stage_activ}, it ``minimizes the cross-coupling amongst actuators and sensors of different legs'' (being orthogonal to each other).
Specific points of interest are:
A number of properties are attributed to the cubic configuration, which have made this configuration widely popular (\cite{geng94_six_degree_of_freed_activ,preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm}):
\begin{itemize}
\item uniform mobility, uniform stiffness, and coupling properties
\item Simple kinematics relationships and dynamical analysis \cite{geng94_six_degree_of_freed_activ}
\item Uniform stiffness in all directions \cite{hanieh03_activ_stewar}
\item Uniform mobility \cite[, chapt.8.5.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}
\item Minimization of the cross coupling between actuators and sensors in other struts \cite{preumont07_six_axis_singl_stage_activ}.
This is attributed to the fact that the struts are orthogonal to each other.
This 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}.
\end{itemize}
In this section:
Such properties are studied to see if they are useful for the nano-hexapod and the associated conditions:
\begin{itemize}
\item Such properties are studied
\item Additional properties interesting for control?
\item It is determined if the cubic architecture is interested for the nano-hexapod
\item The mobility and stiffness properties of the cubic configuration are studied in Section \ref{ssec:detail_kinematics_cubic_static}.
\item Dynamical decoupling is studied in Section \ref{ssec:detail_kinematics_cubic_dynamic}
\item Decentralized control, important for the NASS, is studied in Section \ref{ssec:detail_kinematics_decentralized_control}
\end{itemize}
As the cubic architecture has some restrictions on the geometry, alternative designs are proposed in Section \ref{ssec:detail_kinematics_cubic_design}.
The goal is to determine if the cubic architecture is interesting for the nano-hexapod.
\section{Static Properties}
\label{ssec:detail_kinematics_cubic_static}
\paragraph{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 \eqref{eq:detail_kinematics_cubic_s}.
\begin{equation}\label{eq:detail_kinematics_cubic_s}
@ -567,29 +532,23 @@ Coordinates of the cube's vertices relevant for the top joints, expressed with r
\end{equation}
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.6]{figs/detail_kinematics_cubic_schematic_full.png}
\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_cubic_schematic_full.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_schematic_full}Full cube}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.6]{figs/detail_kinematics_cubic_schematic.png}
\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_cubic_schematic.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_schematic}Cube's portion}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.6]{figs/detail_kinematics_cubic_schematic_off_centered.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_schematic_off_centered}Off Centered}
\end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_schematic_cases}Struts are represented un blue. The cube's center by a dot.}
\end{figure}
In that case (top joints at the cube's vertices), a diagonal stiffness matrix is obtained \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 Hc.
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}
@ -602,16 +561,16 @@ Translation stiffness is twice the stiffness of the struts, and rotational stiff
\end{bmatrix}
\end{equation}
But typically, the top joints are not placed at the cube's vertices but on the cube's edges (Figure \ref{fig:detail_kinematics_cubic_schematic}).
But typically, the top joints are not placed at the cube's vertices but anywhere along the cube's edges (Figure \ref{fig:detail_kinematics_cubic_schematic}).
In that case, the location of the top joints can be expressed by \eqref{eq:detail_kinematics_cubic_edges}.
But the computed stiffness matrix is the same \eqref{eq:detail_kinematics_cubic_stiffness}.
\begin{equation}\label{eq:detail_kinematics_cubic_edges}
\bm{b}_i = \tilde{\bm{b}}_i + \alpha \hat{\bm{s}}_i
\end{equation}
But the computed stiffness matrix is the same \eqref{eq:detail_kinematics_cubic_stiffness}.
The Stiffness matrix is diagonal for forces and torques applied on the top platform, but expressed at the center of the cube, and for translations and rotations of the top platform expressed with respect to the cube's center.
The Stiffness matrix is therefore diagonal when the considered \{B\} frame is located at the center of the cube.
for forces and torques applied on the top platform, but expressed at the center of the cube, and for translations and rotations of the top platform expressed with respect to the cube's center.
\begin{itemize}
\item[{$\square$}] Should I introduce the term ``center of stiffness'' here?
@ -621,7 +580,7 @@ The Stiffness matrix is diagonal for forces and torques applied on the top platf
However, as soon as the location of the A and B frames are shifted from the cube's center, off diagonal elements in the stiffness matrix appear.
Let's consider here a vertical shift as shown in Figure \ref{fig:detail_kinematics_cubic_schematic_off_centered}.
Let's consider here a vertical shift as shown in Figure \ref{fig:detail_kinematics_cubic_schematic}.
In that case, the stiffness matrix is \eqref{eq:detail_kinematics_cubic_stiffness_off_centered}.
Off diagonal elements are increasing with the height difference between the cube's center and the considered B frame.
@ -761,7 +720,7 @@ To verify that,
\paragraph{Payload's CoM at the cube's center}
It is therefore natural to try to have the cube's center and the center of mass of the moving part coincide at the same location.
It is therefore natural to try to have the cube's center and the center of mass of the moving part coincide at the same location \cite{li01_simul_fault_vibrat_isolat_point}.
\begin{itemize}
\item CoM at the center of the cube: Figure \ref{fig:detail_kinematics_cubic_centered_payload}
@ -1055,6 +1014,8 @@ Requirements:
\item The nano-hexapod should fit within a cylinder with radius of \(120\,mm\) and with a height of \(95\,mm\).
\item In terms of mobility: uniform mobility in XYZ directions (100um)
\item In terms of stiffness: ??
Having the resonance frequencies well above the maximum rotational velocity of \(2\pi\,\text{rad/s}\) to limit the gyroscopic effects.
Having the resonance below the problematic modes of the micro-station to decouple from the micro-station complex dynamics.
\item In terms of dynamics:
\begin{itemize}
\item be able to apply IFF in a decentralized way with good robustness and performances (good damping of modes)
@ -1062,8 +1023,14 @@ Requirements:
\end{itemize}
\end{itemize}
For the NASS, the payloads can have various inertia, with masses ranging from 1 to 50kg.
The main difficulty for the design optimization of the nano-hexapod, is that the payloads will have various inertia, with masses ranging from 1 to 50kg.
It is therefore not possible to have one geometry that gives good dynamical properties for all the payloads.
It could have been an option to have a cubic architecture as proposed in section \ref{ssec:detail_kinematics_cubic_design}, but having the cube's center 150mm above the top platform would have lead to platforms well exceeding the maximum available size.
In that case, each payload would have to be calibrated in inertia before placing on top of the nano-hexapod, which would require a lot of work from the future users.
Considering the fact that it would not be possible to have the center of mass at the cube's center, the cubic architecture is not of great value here.
\section{Obtained Geometry}
Take both platforms at maximum size.
@ -1071,9 +1038,26 @@ Make reasonable choice (close to the final choice).
Say that it is good enough to make all the calculations.
The geometry will be slightly refined during the detailed mechanical design for several reason: easy of mount, manufacturability, \ldots{}
\begin{itemize}
\item[{$\square$}] Show the obtained geometry and the main parameters.
\end{itemize}
Obtained geometry is shown in Figure \ref{fig:detail_kinematics_nano_hexapod}.
Height between the top plates is 95mm.
Joints are offset by 15mm from the plate surfaces, and are positioned along a circle with radius 120mm for the fixed joints and 110mm for the mobile joints.
The positioning angles (Figure \ref{fig:detail_kinematics_nano_hexapod_top}) are \([255, 285, 15, 45, 135, 165]\) degrees for the top joints and \([220, 320, 340, 80, 100, 200]\) degrees for the bottom joints.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_nano_hexapod_iso.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_nano_hexapod_iso}Isometric view}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_nano_hexapod_top.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_nano_hexapod_top}Top view}
\end{subfigure}
\caption{\label{fig:detail_kinematics_nano_hexapod}Obtained architecture for the Nano Hexapod}
\end{figure}
This geometry will be used for:
\begin{itemize}