tdehaeze/phd-simscape-nano-hexapod
Archived
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Correct one footnote

Browse Source
This commit is contained in:
Thomas Dehaeze 2025-04-03 22:06:03 +02:00
parent 1199f9fb7a
commit 5db04e754f
3 changed files with 16 additions and 54 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

2
simscape-nano-hexapod.org
View File

@ -729,7 +729,7 @@ While the previously derived kinematic relationships are essential for position
As discussed in Section ref:ssec:nhexa_stewart_platform_kinematics, the strut lengths $\bm{\mathcal{L}}$ and the platform pose $\bm{\mathcal{X}}$ are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts. As discussed in Section ref:ssec:nhexa_stewart_platform_kinematics, the strut lengths $\bm{\mathcal{L}}$ and the platform pose $\bm{\mathcal{X}}$ are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts.
By taking the time derivative of the position loop close eqref:eq:nhexa_loop_closure, equation eqref:eq:nhexa_loop_closure_velocity[fn:nhexa_3] is obtained. By taking the time derivative of the position loop close eqref:eq:nhexa_loop_closure, equation eqref:eq:nhexa_loop_closure_velocity is obtained[fn:nhexa_3].
\begin{equation}\label{eq:nhexa_loop_closure_velocity} \begin{equation}\label{eq:nhexa_loop_closure_velocity}
{}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{\bm{a}}_i}_{=0} {}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{\bm{a}}_i}_{=0}

BIN
simscape-nano-hexapod.pdf
View File

Binary file not shown.

68
simscape-nano-hexapod.tex
View File

@ -1,4 +1,4 @@
% Created 2025-03-28 Fri 14:30 % Created 2025-04-03 Thu 22:05
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -8,13 +8,6 @@
\author{Dehaeze Thomas} \author{Dehaeze Thomas}
\date{\today} \date{\today}
\title{Simscape Model - Nano Hexapod} \title{Simscape Model - Nano Hexapod}
\hypersetup{
pdfauthor={Dehaeze Thomas},
pdftitle={Simscape Model - Nano Hexapod},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 29.4 (Org mode 9.6)},
pdflang={English}}
\usepackage{biblatex} \usepackage{biblatex}
\begin{document} \begin{document}
@ -23,7 +16,6 @@
\tableofcontents \tableofcontents
\clearpage \clearpage
Building upon the validated multi-body model of the micro-station presented in previous sections, this section focuses on the development and integration of an active vibration platform model. Building upon the validated multi-body model of the micro-station presented in previous sections, this section focuses on the development and integration of an active vibration platform model.
A review of existing active vibration platforms is given in Section \ref{sec:nhexa_platform_review}, leading to the selection of the Stewart platform architecture. A review of existing active vibration platforms is given in Section \ref{sec:nhexa_platform_review}, leading to the selection of the Stewart platform architecture.
@ -36,7 +28,6 @@ The control of the Stewart platform introduces additional complexity due to its
Section \ref{sec:nhexa_control} explores how the High Authority Control/Low Authority Control (HAC-LAC) strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform. Section \ref{sec:nhexa_control} explores how the High Authority Control/Low Authority Control (HAC-LAC) strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform.
This adaptation requires fundamental decisions regarding both the control architecture (centralized versus decentralized) and the control frame (Cartesian versus strut space). This adaptation requires fundamental decisions regarding both the control architecture (centralized versus decentralized) and the control frame (Cartesian versus strut space).
Through careful analysis of system interactions and plant characteristics in different frames, a control architecture combining decentralized Integral Force Feedback for active damping with a centralized high authority controller for positioning was developed, with both controllers implemented in the frame of the struts. Through careful analysis of system interactions and plant characteristics in different frames, a control architecture combining decentralized Integral Force Feedback for active damping with a centralized high authority controller for positioning was developed, with both controllers implemented in the frame of the struts.
\chapter{Active Vibration Platforms} \chapter{Active Vibration Platforms}
\label{sec:nhexa_platform_review} \label{sec:nhexa_platform_review}
The conceptual phase started with the use of simplified models, such as uniaxial and three-degree-of-freedom rotating systems. The conceptual phase started with the use of simplified models, such as uniaxial and three-degree-of-freedom rotating systems.
@ -169,7 +160,6 @@ The resulting mechanical structure exhibits complex dynamics with multiple reson
This dynamic complexity poses significant challenges for the design and control of the active platform. This dynamic complexity poses significant challenges for the design and control of the active platform.
The primary control requirements focus on \([D_y,\ D_z,\ R_y]\) motions; however, the continuous rotation of the active platform requires the control of \([D_x,\ D_y,\ D_z,\ R_x,\ R_y]\) in the active platform's reference frame. The primary control requirements focus on \([D_y,\ D_z,\ R_y]\) motions; however, the continuous rotation of the active platform requires the control of \([D_x,\ D_y,\ D_z,\ R_x,\ R_y]\) in the active platform's reference frame.
\section{Active Vibration Platform} \section{Active Vibration Platform}
\label{ssec:nhexa_active_platforms} \label{ssec:nhexa_active_platforms}
@ -231,7 +221,6 @@ Furthermore, the successful implementation of Integral Force Feedback (IFF) cont
\end{subfigure} \end{subfigure}
\caption{\label{fig:nhexa_stewart_examples}Two examples of Stewart platform. A Stewart platform based on piezoelectric stack actuators and used for nano-positioning is shown in (\subref{fig:nhexa_stewart_piezo_furutani}) \cite{furutani04_nanom_cuttin_machin_using_stewar}. A Stewart platform based on voice coil actuators and used for vibration isolation is shown in (\subref{fig:nhexa_stewart_vc_preumont}) \cite{preumont07_six_axis_singl_stage_activ,preumont18_vibrat_contr_activ_struc_fourt_edition}} \caption{\label{fig:nhexa_stewart_examples}Two examples of Stewart platform. A Stewart platform based on piezoelectric stack actuators and used for nano-positioning is shown in (\subref{fig:nhexa_stewart_piezo_furutani}) \cite{furutani04_nanom_cuttin_machin_using_stewar}. A Stewart platform based on voice coil actuators and used for vibration isolation is shown in (\subref{fig:nhexa_stewart_vc_preumont}) \cite{preumont07_six_axis_singl_stage_activ,preumont18_vibrat_contr_activ_struc_fourt_edition}}
\end{figure} \end{figure}
\chapter{The Stewart platform} \chapter{The Stewart platform}
\label{sec:nhexa_stewart_platform} \label{sec:nhexa_stewart_platform}
The Stewart platform, first introduced by Stewart in 1965 \cite{stewart65_platf_with_six_degrees_freed} for flight simulation applications, represents a significant milestone in parallel manipulator design. The Stewart platform, first introduced by Stewart in 1965 \cite{stewart65_platf_with_six_degrees_freed} for flight simulation applications, represents a significant milestone in parallel manipulator design.
@ -288,10 +277,9 @@ This is summarized in Figure \ref{fig:nhexa_stewart_notations}.
\includegraphics[scale=1]{figs/nhexa_stewart_notations.png} \includegraphics[scale=1]{figs/nhexa_stewart_notations.png}
\caption{\label{fig:nhexa_stewart_notations}Frame and key notations for the Stewart platform} \caption{\label{fig:nhexa_stewart_notations}Frame and key notations for the Stewart platform}
\end{figure} \end{figure}
\section{Kinematic Analysis} \section{Kinematic Analysis}
\label{ssec:nhexa_stewart_platform_kinematics} \label{ssec:nhexa_stewart_platform_kinematics}
\paragraph{Loop Closure} \subsubsection{Loop Closure}
The foundation of the kinematic analysis lies in the geometric constraints imposed by each strut, which can be expressed using loop closure equations. The foundation of the kinematic analysis lies in the geometric constraints imposed by each strut, which can be expressed using loop closure equations.
For each strut \(i\) (illustrated in Figure \ref{fig:nhexa_stewart_loop_closure}), the loop closure equation \eqref{eq:nhexa_loop_closure} can be written. For each strut \(i\) (illustrated in Figure \ref{fig:nhexa_stewart_loop_closure}), the loop closure equation \eqref{eq:nhexa_loop_closure} can be written.
@ -307,8 +295,7 @@ This equation links the pose\footnote{The \emph{pose} represents the position an
\includegraphics[scale=1]{figs/nhexa_stewart_loop_closure.png} \includegraphics[scale=1]{figs/nhexa_stewart_loop_closure.png}
\caption{\label{fig:nhexa_stewart_loop_closure}Notations to compute the kinematic loop closure} \caption{\label{fig:nhexa_stewart_loop_closure}Notations to compute the kinematic loop closure}
\end{figure} \end{figure}
\subsubsection{Inverse Kinematics}
\paragraph{Inverse Kinematics}
The inverse kinematic problem involves determining the required strut lengths \(\bm{\mathcal{L}} = \left[ l_1, l_2, \ldots, l_6 \right]^T\) for a desired platform pose \(\bm{\mathcal{X}}\) (i.e. position \({}^A\bm{P}\) and orientation \({}^A\bm{R}_B\)). The inverse kinematic problem involves determining the required strut lengths \(\bm{\mathcal{L}} = \left[ l_1, l_2, \ldots, l_6 \right]^T\) for a desired platform pose \(\bm{\mathcal{X}}\) (i.e. position \({}^A\bm{P}\) and orientation \({}^A\bm{R}_B\)).
This problem can be solved analytically using the loop closure equations \eqref{eq:nhexa_loop_closure}. This problem can be solved analytically using the loop closure equations \eqref{eq:nhexa_loop_closure}.
@ -320,8 +307,7 @@ The obtained strut lengths are given by \eqref{eq:nhexa_inverse_kinematics}.
If the position and orientation of the platform lie in the feasible workspace, the solution is unique. If the position and orientation of the platform lie in the feasible workspace, the solution is unique.
While configurations outside this workspace yield complex numbers, this only becomes relevant for large displacements that far exceed the nano-hexapod's operating range. While configurations outside this workspace yield complex numbers, this only becomes relevant for large displacements that far exceed the nano-hexapod's operating range.
\subsubsection{Forward Kinematics}
\paragraph{Forward Kinematics}
The forward kinematic problem seeks to determine the platform pose \(\bm{\mathcal{X}}\) given a set of strut lengths \(\bm{\mathcal{L}}\). The forward kinematic problem seeks to determine the platform pose \(\bm{\mathcal{X}}\) given a set of strut lengths \(\bm{\mathcal{L}}\).
Unlike inverse kinematics, this presents a significant challenge because it requires solving a system of nonlinear equations. Unlike inverse kinematics, this presents a significant challenge because it requires solving a system of nonlinear equations.
@ -329,17 +315,15 @@ Although various numerical methods exist for solving this problem, they can be c
For the nano-hexapod application, where displacements are typically small, an approximate solution based on linearization around the operating point provides a practical alternative. For the nano-hexapod application, where displacements are typically small, an approximate solution based on linearization around the operating point provides a practical alternative.
This approximation, which is developed in subsequent sections through the Jacobian matrix analysis, is particularly useful for real-time control applications. This approximation, which is developed in subsequent sections through the Jacobian matrix analysis, is particularly useful for real-time control applications.
\section{The Jacobian Matrix} \section{The Jacobian Matrix}
\label{ssec:nhexa_stewart_platform_jacobian} \label{ssec:nhexa_stewart_platform_jacobian}
The Jacobian matrix plays a central role in analyzing the Stewart platform's behavior, providing a linear mapping between the platform and actuator velocities. The Jacobian matrix plays a central role in analyzing the Stewart platform's behavior, providing a linear mapping between the platform and actuator velocities.
While the previously derived kinematic relationships are essential for position analysis, the Jacobian enables velocity analysis and forms the foundation for both static and dynamic studies. While the previously derived kinematic relationships are essential for position analysis, the Jacobian enables velocity analysis and forms the foundation for both static and dynamic studies.
\paragraph{Jacobian Computation} \subsubsection{Jacobian Computation}
As discussed in Section \ref{ssec:nhexa_stewart_platform_kinematics}, the strut lengths \(\bm{\mathcal{L}}\) and the platform pose \(\bm{\mathcal{X}}\) are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts. As discussed in Section \ref{ssec:nhexa_stewart_platform_kinematics}, the strut lengths \(\bm{\mathcal{L}}\) and the platform pose \(\bm{\mathcal{X}}\) are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts.
By taking the time derivative of the position loop close \eqref{eq:nhexa_loop_closure}, equation \eqref{eq:nhexa_loop_closure_velocity}\footnote{Such equation is called the \emph{velocity loop closure}} is obtained. By taking the time derivative of the position loop close \eqref{eq:nhexa_loop_closure}, equation \eqref{eq:nhexa_loop_closure_velocity} is obtained\footnote{Such equation is called the \emph{velocity loop closure}}.
\begin{equation}\label{eq:nhexa_loop_closure_velocity} \begin{equation}\label{eq:nhexa_loop_closure_velocity}
{}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{\bm{a}}_i}_{=0} {}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{\bm{a}}_i}_{=0}
@ -378,8 +362,7 @@ The matrix \(\bm{J}\) is called the Jacobian matrix and is defined by \eqref{eq:
Therefore, the Jacobian matrix \(\bm{J}\) links the rate of change of the strut length to the velocity and angular velocity of the top platform with respect to the fixed base through a set of linear equations. Therefore, the Jacobian matrix \(\bm{J}\) links the rate of change of the strut length to the velocity and angular velocity of the top platform with respect to the fixed base through a set of linear equations.
However, \(\bm{J}\) needs to be recomputed for every Stewart platform pose because it depends on the actual pose of the manipulator. However, \(\bm{J}\) needs to be recomputed for every Stewart platform pose because it depends on the actual pose of the manipulator.
\subsubsection{Approximate solution to the Forward and Inverse Kinematic problems}
\paragraph{Approximate solution to the Forward and Inverse Kinematic problems}
For small displacements \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T\) around an operating point \(\bm{\mathcal{X}}_0\) (for which the Jacobian was computed), the associated joint displacement \(\delta\bm{\mathcal{L}} = [\delta l_1,\,\delta l_2,\,\delta l_3,\,\delta l_4,\,\delta l_5,\,\delta l_6]^T\) can be computed using the Jacobian \eqref{eq:nhexa_inverse_kinematics_approximate}. For small displacements \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T\) around an operating point \(\bm{\mathcal{X}}_0\) (for which the Jacobian was computed), the associated joint displacement \(\delta\bm{\mathcal{L}} = [\delta l_1,\,\delta l_2,\,\delta l_3,\,\delta l_4,\,\delta l_5,\,\delta l_6]^T\) can be computed using the Jacobian \eqref{eq:nhexa_inverse_kinematics_approximate}.
@ -395,8 +378,7 @@ Similarly, for small joint displacements \(\delta\bm{\mathcal{L}}\), it is possi
These two relations solve the forward and inverse kinematic problems for small displacement in a \emph{approximate} way. These two relations solve the forward and inverse kinematic problems for small displacement in a \emph{approximate} way.
While this approximation offers limited value for inverse kinematics, which can be solved analytically, it proves particularly useful for the forward kinematic problem where exact analytical solutions are difficult to obtain. While this approximation offers limited value for inverse kinematics, which can be solved analytically, it proves particularly useful for the forward kinematic problem where exact analytical solutions are difficult to obtain.
\subsubsection{Range validity of the approximate inverse kinematics}
\paragraph{Range validity of the approximate inverse kinematics}
The accuracy of the Jacobian-based forward kinematics solution was estimated by a simple analysis. The accuracy of the Jacobian-based forward kinematics solution was estimated by a simple analysis.
For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation \eqref{eq:nhexa_inverse_kinematics}. For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation \eqref{eq:nhexa_inverse_kinematics}.
@ -413,8 +395,7 @@ It can be computed once at the rest position and used for both forward and inver
\includegraphics[scale=1]{figs/nhexa_forward_kinematics_approximate_errors.png} \includegraphics[scale=1]{figs/nhexa_forward_kinematics_approximate_errors.png}
\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,mm\) was used to perform this analysis. \(\epsilon_D\) corresponds to the distance between the true positioin and the estimated position. \(\epsilon_R\) corresponds to the angular motion between the true orientation and the estimated orientation.} \caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,mm\) was used to perform this analysis. \(\epsilon_D\) corresponds to the distance between the true positioin and the estimated position. \(\epsilon_R\) corresponds to the angular motion between the true orientation and the estimated orientation.}
\end{figure} \end{figure}
\subsubsection{Static Forces}
\paragraph{Static Forces}
The static force analysis of the Stewart platform can be performed using the principle of virtual work. The static force analysis of the Stewart platform can be performed using the principle of virtual work.
This principle states that for a system in static equilibrium, the total virtual work of all forces acting on the system must be zero for any virtual displacement compatible with the system's constraints. This principle states that for a system in static equilibrium, the total virtual work of all forces acting on the system must be zero for any virtual displacement compatible with the system's constraints.
@ -443,7 +424,6 @@ Because this equation must hold for any virtual displacement \(\delta \bm{\mathc
\end{equation} \end{equation}
These equations establish that the transpose of the Jacobian matrix maps actuator forces to platform forces and torques, while its inverse transpose maps platform forces and torques to required actuator forces. These equations establish that the transpose of the Jacobian matrix maps actuator forces to platform forces and torques, while its inverse transpose maps platform forces and torques to required actuator forces.
\section{Static Analysis} \section{Static Analysis}
\label{ssec:nhexa_stewart_platform_static} \label{ssec:nhexa_stewart_platform_static}
@ -485,7 +465,6 @@ However, the initial geometric configuration significantly affects the overall s
The relationship between maximum stroke and stiffness presents another important design consideration. The relationship between maximum stroke and stiffness presents another important design consideration.
As both parameters are influenced by the geometric configuration, their optimization involves inherent trade-offs that must be carefully balanced based on the application requirements. As both parameters are influenced by the geometric configuration, their optimization involves inherent trade-offs that must be carefully balanced based on the application requirements.
The optimization of this configuration to achieve the desired stiffness while having sufficient stroke will be addressed during the detailed design phase. The optimization of this configuration to achieve the desired stiffness while having sufficient stroke will be addressed during the detailed design phase.
\section{Dynamical Analysis} \section{Dynamical Analysis}
\label{ssec:nhexa_stewart_platform_dynamics} \label{ssec:nhexa_stewart_platform_dynamics}
@ -531,7 +510,6 @@ Several factors can significantly increase the model complexity, such as:
\end{itemize} \end{itemize}
These additional effects render analytical modeling impractical for complete system analysis. These additional effects render analytical modeling impractical for complete system analysis.
\section*{Conclusion} \section*{Conclusion}
The fundamental characteristics of the Stewart platform have been analyzed in this chapter. The fundamental characteristics of the Stewart platform have been analyzed in this chapter.
Essential kinematic relationships were developed through loop closure equations, from which both exact and approximate solutions for the inverse and forward kinematic problems were derived. Essential kinematic relationships were developed through loop closure equations, from which both exact and approximate solutions for the inverse and forward kinematic problems were derived.
@ -543,7 +521,6 @@ This will be performed in the next section using a multi-body model.
All these characteristics (maneuverability, stiffness, dynamics, etc.) are fundamentally determined by the platform's geometry. All these characteristics (maneuverability, stiffness, dynamics, etc.) are fundamentally determined by the platform's geometry.
While a reasonable geometric configuration will be used to validate the NASS during the conceptual phase, the optimization of these geometric parameters will be explored during the detailed design phase. While a reasonable geometric configuration will be used to validate the NASS during the conceptual phase, the optimization of these geometric parameters will be explored during the detailed design phase.
\chapter{Multi-Body Model} \chapter{Multi-Body Model}
\label{sec:nhexa_model} \label{sec:nhexa_model}
The dynamic modeling of Stewart platforms has traditionally relied on analytical approaches. The dynamic modeling of Stewart platforms has traditionally relied on analytical approaches.
@ -557,7 +534,7 @@ The model is then validated through comparison with the analytical equations in
Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section \ref{ssec:nhexa_model_dynamics}). Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section \ref{ssec:nhexa_model_dynamics}).
\section{Model Definition} \section{Model Definition}
\label{ssec:nhexa_model_def} \label{ssec:nhexa_model_def}
\paragraph{Geometry} \subsubsection{Geometry}
The Stewart platform's geometry is defined by two principal coordinate frames (Figure \ref{fig:nhexa_stewart_model_def}): a fixed base frame \(\{F\}\) and a moving platform frame \(\{M\}\). The Stewart platform's geometry is defined by two principal coordinate frames (Figure \ref{fig:nhexa_stewart_model_def}): a fixed base frame \(\{F\}\) and a moving platform frame \(\{M\}\).
The joints connecting the actuators to these frames are located at positions \({}^F\bm{a}_i\) and \({}^M\bm{b}_i\) respectively. The joints connecting the actuators to these frames are located at positions \({}^F\bm{a}_i\) and \({}^M\bm{b}_i\) respectively.
@ -600,15 +577,13 @@ From these parameters, key kinematic properties can be derived: the strut orient
\captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]} \captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]}
\end{scriptsize} \end{scriptsize}
\end{minipage} \end{minipage}
\subsubsection{Inertia of Plates}
\paragraph{Inertia of Plates}
The fixed base and moving platform were modeled as solid cylindrical bodies. The fixed base and moving platform were modeled as solid cylindrical bodies.
The base platform was characterized by a radius of \(120\,mm\) and thickness of \(15\,mm\), matching the dimensions of the micro-hexapod's top platform. The base platform was characterized by a radius of \(120\,mm\) and thickness of \(15\,mm\), matching the dimensions of the micro-hexapod's top platform.
The moving platform was similarly modeled with a radius of \(110\,mm\) and thickness of \(15\,mm\). The moving platform was similarly modeled with a radius of \(110\,mm\) and thickness of \(15\,mm\).
Both platforms were assigned a mass of \(5\,kg\). Both platforms were assigned a mass of \(5\,kg\).
\subsubsection{Joints}
\paragraph{Joints}
The platform's joints play a crucial role in its dynamic behavior. The platform's joints play a crucial role in its dynamic behavior.
At both the upper and lower connection points, various degrees of freedom can be modeled, including universal joints, spherical joints, and configurations with additional axial and lateral stiffness components. At both the upper and lower connection points, various degrees of freedom can be modeled, including universal joints, spherical joints, and configurations with additional axial and lateral stiffness components.
@ -616,8 +591,7 @@ For each degree of freedom, stiffness characteristics can be incorporated into t
In the conceptual design phase, a simplified joint configuration is employed: the bottom joints are modeled as two-degree-of-freedom universal joints, while the top joints are represented as three-degree-of-freedom spherical joints. In the conceptual design phase, a simplified joint configuration is employed: the bottom joints are modeled as two-degree-of-freedom universal joints, while the top joints are represented as three-degree-of-freedom spherical joints.
These joints are considered massless and exhibit no stiffness along their degrees of freedom. These joints are considered massless and exhibit no stiffness along their degrees of freedom.
\subsubsection{Actuators}
\paragraph{Actuators}
The actuator model comprises several key elements (Figure \ref{fig:nhexa_actuator_model}). The actuator model comprises several key elements (Figure \ref{fig:nhexa_actuator_model}).
At its core, each actuator is modeled as a prismatic joint with internal stiffness \(k_a\) and damping \(c_a\), driven by a force source \(f\). At its core, each actuator is modeled as a prismatic joint with internal stiffness \(k_a\) and damping \(c_a\), driven by a force source \(f\).
@ -650,7 +624,6 @@ This modular approach to actuator modeling allows for future refinements as the
\captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters} \captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters}
\end{scriptsize} \end{scriptsize}
\end{minipage} \end{minipage}
\section{Validation of the multi-body model} \section{Validation of the multi-body model}
\label{ssec:nhexa_model_validation} \label{ssec:nhexa_model_validation}
@ -700,7 +673,6 @@ The close agreement between both approaches across the frequency spectrum valida
\includegraphics[scale=1]{figs/nhexa_comp_multi_body_analytical.png} \includegraphics[scale=1]{figs/nhexa_comp_multi_body_analytical.png}
\caption{\label{fig:nhexa_comp_multi_body_analytical}Comparison of the analytical transfer functions and the multi-body model} \caption{\label{fig:nhexa_comp_multi_body_analytical}Comparison of the analytical transfer functions and the multi-body model}
\end{figure} \end{figure}
\section{Nano Hexapod Dynamics} \section{Nano Hexapod Dynamics}
\label{ssec:nhexa_model_dynamics} \label{ssec:nhexa_model_dynamics}
@ -737,7 +709,6 @@ The inclusion of parallel stiffness introduces an additional complex conjugate z
\end{subfigure} \end{subfigure}
\caption{\label{fig:nhexa_multi_body_plant}Bode plot of the transfer functions computed from the nano-hexapod multi-body model} \caption{\label{fig:nhexa_multi_body_plant}Bode plot of the transfer functions computed from the nano-hexapod multi-body model}
\end{figure} \end{figure}
\section*{Conclusion} \section*{Conclusion}
The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the nano-hexapod system. The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the nano-hexapod system.
Through comparison with analytical solutions in a simplified configuration, the model's accuracy has been validated, demonstrating its ability to capture the essential dynamic behavior of the Stewart platform. Through comparison with analytical solutions in a simplified configuration, the model's accuracy has been validated, demonstrating its ability to capture the essential dynamic behavior of the Stewart platform.
@ -746,7 +717,6 @@ A key advantage of this modeling approach lies in its flexibility for future ref
While the current implementation employs idealized joints for the conceptual design phase, the framework readily accommodates the incorporation of joint stiffness and other non-ideal effects. While the current implementation employs idealized joints for the conceptual design phase, the framework readily accommodates the incorporation of joint stiffness and other non-ideal effects.
The joint stiffness, which is known to impact the performance of decentralized IFF control strategy \cite{preumont07_six_axis_singl_stage_activ}, will be studied and optimized during the detailed design phase. The joint stiffness, which is known to impact the performance of decentralized IFF control strategy \cite{preumont07_six_axis_singl_stage_activ}, will be studied and optimized during the detailed design phase.
The validated multi-body model will serve as a valuable tool for predicting system behavior and evaluating control performance throughout the design process. The validated multi-body model will serve as a valuable tool for predicting system behavior and evaluating control performance throughout the design process.
\chapter{Control of Stewart Platforms} \chapter{Control of Stewart Platforms}
\label{sec:nhexa_control} \label{sec:nhexa_control}
The control of Stewart platforms presents distinct challenges compared to the uniaxial model due to their multi-input multi-output nature. The control of Stewart platforms presents distinct challenges compared to the uniaxial model due to their multi-input multi-output nature.
@ -783,14 +753,12 @@ In the context of the nano-hexapod, two distinct control strategies were examine
\includegraphics[scale=1]{figs/nhexa_stewart_decentralized_control.png} \includegraphics[scale=1]{figs/nhexa_stewart_decentralized_control.png}
\caption{\label{fig:nhexa_stewart_decentralized_control}Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.} \caption{\label{fig:nhexa_stewart_decentralized_control}Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.}
\end{figure} \end{figure}
\section{Choice of the Control Space} \section{Choice of the Control Space}
\label{ssec:nhexa_control_space} \label{ssec:nhexa_control_space}
When controlling a Stewart platform using external metrology that measures the pose of frame \(\{B\}\) with respect to \(\{A\}\), denoted as \(\bm{\mathcal{X}}\), the control architecture can be implemented in either Cartesian or strut space. When controlling a Stewart platform using external metrology that measures the pose of frame \(\{B\}\) with respect to \(\{A\}\), denoted as \(\bm{\mathcal{X}}\), the control architecture can be implemented in either Cartesian or strut space.
This choice affects both the control design and the obtained performance. This choice affects both the control design and the obtained performance.
\subsubsection{Control in the Strut space}
\paragraph{Control in the Strut space}
In this approach, as illustrated in Figure \ref{fig:nhexa_control_strut}, the control is performed in the space of the struts. In this approach, as illustrated in Figure \ref{fig:nhexa_control_strut}, the control is performed in the space of the struts.
The Jacobian matrix is used to solve the inverse kinematics in real-time by mapping position errors from Cartesian space \(\bm{\epsilon}_{\mathcal{X}}\) to strut space \(\bm{\epsilon}_{\mathcal{L}}\). The Jacobian matrix is used to solve the inverse kinematics in real-time by mapping position errors from Cartesian space \(\bm{\epsilon}_{\mathcal{X}}\) to strut space \(\bm{\epsilon}_{\mathcal{L}}\).
@ -818,8 +786,7 @@ Furthermore, at low frequencies, the plant exhibits good decoupling between the
\end{subfigure} \end{subfigure}
\caption{\label{fig:nhexa_control_frame}Two control strategies} \caption{\label{fig:nhexa_control_frame}Two control strategies}
\end{figure} \end{figure}
\subsubsection{Control in Cartesian Space}
\paragraph{Control in Cartesian Space}
Alternatively, control can be implemented directly in Cartesian space, as illustrated in Figure \ref{fig:nhexa_control_cartesian}. Alternatively, control can be implemented directly in Cartesian space, as illustrated in Figure \ref{fig:nhexa_control_cartesian}.
Here, the controller processes Cartesian errors \(\bm{\epsilon}_{\mathcal{X}}\) to generate forces and torques \(\bm{\mathcal{F}}\), which are then mapped to actuator forces using the transpose of the inverse Jacobian matrix \eqref{eq:nhexa_jacobian_forces}. Here, the controller processes Cartesian errors \(\bm{\epsilon}_{\mathcal{X}}\) to generate forces and torques \(\bm{\mathcal{F}}\), which are then mapped to actuator forces using the transpose of the inverse Jacobian matrix \eqref{eq:nhexa_jacobian_forces}.
@ -849,7 +816,6 @@ More sophisticated control strategies will be explored during the detailed desig
\end{subfigure} \end{subfigure}
\caption{\label{fig:nhexa_plant_frame}Bode plot of the transfer functions computed from the nano-hexapod multi-body model} \caption{\label{fig:nhexa_plant_frame}Bode plot of the transfer functions computed from the nano-hexapod multi-body model}
\end{figure} \end{figure}
\section{Active Damping with Decentralized IFF} \section{Active Damping with Decentralized IFF}
\label{ssec:nhexa_control_iff} \label{ssec:nhexa_control_iff}
@ -899,7 +865,6 @@ This high gain, combined with the bounded phase, enables effective damping of th
\end{subfigure} \end{subfigure}
\caption{\label{fig:nhexa_decentralized_iff_results}Decentralized IFF} \caption{\label{fig:nhexa_decentralized_iff_results}Decentralized IFF}
\end{figure} \end{figure}
\section{MIMO High-Authority Control - Low-Authority Control} \section{MIMO High-Authority Control - Low-Authority Control}
\label{ssec:nhexa_control_hac_lac} \label{ssec:nhexa_control_hac_lac}
@ -970,7 +935,6 @@ Additionally, the distance of the loci from the \(-1\) point provides informatio
\end{subfigure} \end{subfigure}
\caption{\label{fig:nhexa_decentralized_hac_iff_results}Decentralized HAC-IFF. Loop gain (\subref{fig:nhexa_decentralized_hac_iff_loop_gain}) is used for the design of the controller and to estimate the disturbance rejection performances. Characteristic Loci (\subref{fig:nhexa_decentralized_hac_iff_root_locus}) is used to verify the stability and robustness of the feedback loop.} \caption{\label{fig:nhexa_decentralized_hac_iff_results}Decentralized HAC-IFF. Loop gain (\subref{fig:nhexa_decentralized_hac_iff_loop_gain}) is used for the design of the controller and to estimate the disturbance rejection performances. Characteristic Loci (\subref{fig:nhexa_decentralized_hac_iff_root_locus}) is used to verify the stability and robustness of the feedback loop.}
\end{figure} \end{figure}
\section*{Conclusion} \section*{Conclusion}
The control architecture developed for the uniaxial and the rotating models was adapted for the Stewart platform. The control architecture developed for the uniaxial and the rotating models was adapted for the Stewart platform.
@ -981,7 +945,6 @@ The HAC-LAC strategy was then implemented.
The inner loop implements decentralized Integral Force Feedback for active damping. The inner loop implements decentralized Integral Force Feedback for active damping.
The collocated nature of the force sensors ensures stability despite strong coupling between struts at resonance frequencies, enabling effective damping of structural modes. The collocated nature of the force sensors ensures stability despite strong coupling between struts at resonance frequencies, enabling effective damping of structural modes.
The outer loop implements High Authority Control, enabling precise positioning of the mobile platform. The outer loop implements High Authority Control, enabling precise positioning of the mobile platform.
\chapter*{Conclusion} \chapter*{Conclusion}
\label{sec:nhexa_conclusion} \label{sec:nhexa_conclusion}
@ -1002,6 +965,5 @@ Although the coupled dynamics of the system suggest the potential benefit of adv
This approach combines decentralized Integral Force Feedback for active damping with High Authority Control for positioning, which was implemented in the strut space to leverage the natural decoupling observed at low frequencies. This approach combines decentralized Integral Force Feedback for active damping with High Authority Control for positioning, which was implemented in the strut space to leverage the natural decoupling observed at low frequencies.
This study establishes the theoretical framework necessary for the subsequent development and validation of the NASS. This study establishes the theoretical framework necessary for the subsequent development and validation of the NASS.
\printbibliography[heading=bibintoc,title={Bibliography}] \printbibliography[heading=bibintoc,title={Bibliography}]
\end{document} \end{document}