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Add recommendation for geometry

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Thomas Dehaeze
2025-12-02 16:29:00 +01:00
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delta-robot.tex
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@@ -1,4 +1,4 @@
% Created 2025-12-02 Tue 16:08
% Created 2025-12-02 Tue 16:28
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@@ -22,14 +22,15 @@
\tableofcontents
\clearpage
\chapter{The Delta Robot Kinematics}
\label{sec:delta_robot_kinematics}
\chapter{Studied Geometry}
\section{Studied Geometry}
The Delta Robot geometry is defined as shown in Figure \ref{fig:delta_robot_schematic}.
The geometry is fully defined by three parameters:
\begin{itemize}
\item \texttt{d}: Cube's size (i.e., the length of the cube edge)
\item \texttt{a}: Distance from cube's vertex to top flexible joint
\item \texttt{b}: Distance from cube's vertex to top flexible joint
\item \texttt{L}: Distance between two flexible joints (i.e., the length of the struts)
\end{itemize}
@@ -98,7 +99,7 @@ L = 50e-3; % Length of the struts [m]
\includegraphics[scale=1]{figs/delta_robot_architecture_top.png}
\caption{\label{fig:delta_robot_architecture_top}Delta Robot Architecture - Top View}
\end{figure}
\chapter{Kinematics: Jacobian Matrix and Mobility}
\section{Kinematics: Jacobian Matrix and Mobility}
There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\);
@@ -156,7 +157,7 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um
\includegraphics[scale=1]{figs/delta_robot_2d_workspace_optimal.png}
\caption{\label{fig:delta_robot_2d_workspace_optimal}2D mobility for the optimal Rz angle}
\end{figure}
\chapter{Kinematics: Degrees of Freedom}
\section{Kinematics: Degrees of Freedom}
In the perfect case (flexible joints having no stiffness in bending, and infinite stiffness in torsion and in the axial direction), the top platform is allowed to move only in the X, Y and Z directions while the three rotations are fixed.
In order to have some compliance in rotation, the flexible joints need to have some compliance in torsion \textbf{and} in the axial direction.
@@ -203,7 +204,7 @@ In that case we get some compliance in rotation.
So it is a combination of axial and torsion stiffness that gives some rotational stiffness of the top platform.
Therefore, to model some compliance of the top platform in rotation, both the axial compliance and the torsional compliance of the flexible joints should be considered.
\chapter{Kinematics: Number of modes}
\section{Kinematics: Number of modes}
In the perfect condition (i.e. infinite stiffness in torsion and in compression of the flexible joints), the system has 6 states (i.e. 3 modes, one for each DoF: X, Y and Z).
@@ -233,7 +234,7 @@ The goal is to extract specifications for the flexible joints of the six struts,
The cube's edge length is equal to 50mm, the distance between cube's vertices and top joints is 20mm and the length of the struts (i.e. the distance between the two flexible joints of the same strut) is 50mm.
The actuator stiffness is \(1\,N/\mu m\).
The obtained geometry is shown in Figure [].
The obtained geometry is shown in Figure \ref{fig:delta_robot_studied_geometry}.
\begin{figure}[htbp]
\centering
@@ -372,8 +373,10 @@ Shear & Can limit the stiffness of the system & As high as possible (less import
\end{table}
\chapter{Effect of the Geometry}
\label{sec:delta_robot_flexible_geometry}
\section{Effect of cube's size}
Let's choose reasonable values for the flexible joints:
In this section, the effect of the geometry on the system properties are studied.
The goal is to better understand the different trade-offs, and to extract specifications in terms of the Delta Robot geometry.
Reasonable values for the flexible joints are taken:
\begin{itemize}
\item Bending stiffness of 50Nm/rad
\item Torsional stiffness of 500Nm/rad
@@ -381,16 +384,34 @@ Let's choose reasonable values for the flexible joints:
\item Shear stiffness of 100N/um
\end{itemize}
And we see the effect of changing the cube's size.
\subsection{Effect on the plant dynamics}
The effect of the following geometrical features are studied:
\begin{itemize}
\item[{$\square$}] \textbf{Understand why such different dynamics between 3dof\_a joints and 6dof joints with very high shear stiffnesses}
\item The cube's size in Section \ref{ssec:delta_robot_flexible_geometry_cube_size}
\item The strut length in Section \ref{ssec:delta_robot_flexible_geometry_strut_length}
\item The location of the payload's Center of Mass with respect to the cube's center in Section \ref{ssec:delta_robot_flexible_geometry_com}
\end{itemize}
\section{Effect of cube's size}
\label{ssec:delta_robot_flexible_geometry_cube_size}
\subsection{Obtained geometries}
The cube size is varied from 10mm (Figure \ref{fig:delta_robot_cube_size_small}) to 100mm (Figure \ref{fig:delta_robot_cube_size_large}) to study the effect on the system dynamics.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/delta_robot_cube_size_small.png}
\caption{\label{fig:delta_robot_cube_size_small}Obtained Delta Robot for a cube's size of 10mm}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/delta_robot_cube_size_large.png}
\caption{\label{fig:delta_robot_cube_size_large}Obtained Delta Robot for a cube's size of 100mm}
\end{figure}
\subsection{Effect on the plant dynamics}
The effect of the cube's size on the plant dynamics is shown in Figure \ref{fig:delta_robot_cube_size_plant_dynamics}:
\begin{itemize}
\item coupling decreases with the cube's size
\item coupling decreases with the cube's size (probably because of the reduced effect of the flexible joints' bending stiffness)
\item one resonance frequency increases with the cube's size (resonances in rotation), which may be beneficial from a control point of view
\item coupling at the main resonance varies with the cube's size, but it may also depend on the relative position between the CoM and the cube's center
\end{itemize}
@@ -404,14 +425,16 @@ The effect of the cube's size on the plant dynamics is shown in Figure \ref{fig:
As shown in Figure \ref{fig:delta_robot_cube_size_compliance_rotation}, the stiffness of the delta robot in rotation increases with the cube's size.
Therefore, if possible the cube's size should be increased.
With a cube size of 50mm, the resonance frequency is already above 1kHz with seems reasonable.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/delta_robot_cube_size_compliance_rotation.png}
\caption{\label{fig:delta_robot_cube_size_compliance_rotation}Effect of the cube's size on the rotational compliance of the top platform}
\end{figure}
With a cube size of 50mm, the resonance frequency is already above 1kHz with seems reasonable.
\section{Effect of the strut length}
\label{ssec:delta_robot_flexible_geometry_strut_length}
Let's choose reasonable values for the flexible joints:
\begin{itemize}
\item Bending stiffness of 50Nm/rad
@@ -447,7 +470,6 @@ Probably: when pushing with one actuator, it induces some rotation of the struts
This rotation is proportional to the strut length.
Then, this rotation, combined with the limited compliance in bending of the flexible joints induces some force applied on the other actuators, hence the coupling.
This is similar to what was observed when varying the bending stiffness of the flexible joints: the coupling was increased with an increased of the bending stiffness (See Figure \ref{fig:delta_robot_bending_stiffness_couplign})
\textbf{So we should also observed a decrease of the coupling when decreasing the bending stiffness of the actuators}
\end{itemize}
But even with relatively short struts (20mm and above), the low frequency decoupling is already around two orders of magnitude, which is enough from a control point of view.
@@ -459,6 +481,7 @@ So, the struts length can be optimized to not decrease too much the stiffness of
\caption{\label{fig:delta_robot_strut_length_plant_dynamics}Effect of the Strut length on the plant dynamics}
\end{figure}
\section{Having the Center of Mass at the cube's center}
\label{ssec:delta_robot_flexible_geometry_com}
To make things easier, we take a top platform with no mass, mass-less struts, and we put a payload on top of the platform.
@@ -472,6 +495,17 @@ But how sensitive this decoupling is to the exact position of the CoM still need
\caption{\label{fig:delta_robot_CoM_pos_effect_plant}Effect of the payload's Center of Mass position with respect to the cube's size on the plant dynamics}
\end{figure}
\section{Conclusion}
\begin{table}[htbp]
\caption{\label{tab:delta_robot_geometry_recommendations}Recommendations for the Delta Robot Geometry}
\centering
\begin{tabular}{lll}
Geometrical feature & Effect & Recommendation\\
\hline
Cube's size \texttt{d} & Increasing the cube's size increases the rotational stiffness & Should be make as large as possible\\
Strut length \texttt{L} & Changes the stiffness and coupling of the system (by changing the effect of the flexible joint bending stiffness) & Trade-off between higher stiffness and lower coupling\\
\end{tabular}
\end{table}
\chapter{Conclusion}
\printbibliography[heading=bibintoc,title={Bibliography}]