Try using eqref
This commit is contained in:
@@ -1,4 +1,4 @@
|
||||
% Created 2025-12-02 Tue 15:00
|
||||
% Created 2025-12-02 Tue 15:22
|
||||
% 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) \cref{eq:detail_kinematics_cubic_s}
|
||||
\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{L}: Distance between two flexible joints (i.e., the length of the struts)
|
||||
\end{itemize}
|
||||
@@ -93,27 +93,29 @@ Let's initialize a Delta Robot architecture, and plot the obtained geometry (Fig
|
||||
\end{figure}
|
||||
\chapter{Kinematics: Jacobian Matrix and Mobility}
|
||||
|
||||
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}
|
||||
\begin{equation}\label{eq:delta_robot_unit_vectors}
|
||||
\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}
|
||||
The Jacobian matrix is defined as shown in \eqref{eq:delta_robot_jacobian}.
|
||||
|
||||
\begin{equation}\label{eq:delta_robot_jacobian}
|
||||
\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}
|
||||
It links the small actuator displacement to the top platform displacement \eqref{eq:delta_robot_inverse_kinematics}.
|
||||
|
||||
\begin{equation}\label{eq:delta_robot_inverse_kinematics}
|
||||
d\mathcal{L} = J d\mathcal{L}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{equation}\label{eq:delta_robot_forward_kinematics}
|
||||
d\mathcal{X} = J^{-1} d\mathcal{L}
|
||||
\end{equation}
|
||||
|
||||
|
||||
Reference in New Issue
Block a user