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Thomas Dehaeze
2025-12-02 15:08:20 +01:00
parent be49c57495
commit 9231f43cbb
5 changed files with 111 additions and 111 deletions
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18
delta-robot.tex
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@@ -1,4 +1,4 @@
% Created 2025-12-02 Tue 14:32
% Created 2025-12-02 Tue 15:00
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@@ -28,7 +28,7 @@ The Delta Robot geometry is defined as shown in Figure \ref{fig:delta_robot_sche
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{d}: Cube's size (i.e., the length of the cube edge) \cref{eq:detail_kinematics_cubic_s}
\item \texttt{a}: 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}
@@ -98,23 +98,23 @@ Jacobian matrix between actuator displacement and top platform displacement.
There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\);
\begin{equation}\label{eq:detail_kinematics_cubic_s}
\hat{\bm{s}}_1 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_2 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_1 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}\quad
\hat{\bm{s}}_2 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix}\quad
\hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
\end{equation}
\begin{equation}
\bm{J} = \begin{bmatrix}
\hat{\bm{s}}_1^T \\ \hat{\bm{s}}_2^T \\ \hat{\bm{s}}_3^T
\end{bmatrix}
\bm{J} = \begin{bmatrix}
\hat{\bm{s}}_1^T \\ \hat{\bm{s}}_2^T \\ \hat{\bm{s}}_3^T
\end{bmatrix}
\end{equation}
\begin{equation}
d\mathcal{L} = J d\mathcal{L}
d\mathcal{L} = J d\mathcal{L}
\end{equation}
\begin{equation}
d\mathcal{X} = J^{-1} d\mathcal{L}
d\mathcal{X} = J^{-1} d\mathcal{L}
\end{equation}
The achievable workspace is a cube whose edge length is equal to the actuator stroke.