# Optimizing Assembly in Wiring Boxes Using API Technology for Digital Twin
## 3 Overview of the System Architecture
### Kinematics of Robotic Arm
#### 3.2.2 Homogeneous and Combined Transformation Matrices  
From each joint \(i\), the transformation matrix \(T_{i}\) is given by:  
Standard DH Transformation Matrices  
\[T_{i}^{standard}=\left[\begin{matrix}\cos(\theta_{i})&-\sin(\theta_{i})\cos( \alpha_{i})&\sin(\theta_{i})\sin(\alpha_{i})&a_{i}\cos(\theta_{i})\\ \sin(\theta_{i})&\cos(\theta_{i})\cos(\alpha_{i})&-\cos(\theta_{i})\cos( \alpha_{i})&a_{i}\sin(\theta_{i})\\ 0&\sin(\alpha_{i})&cos(\alpha_{i})&d_{i}\\ 0&0&0&1\end{matrix}\right] \tag{1}\]  
Figure 1: Technical representation for Ufactory Lite 6: (**a**) a description of the precise dimensions, (**b**) the isometric view of the robot [17].  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Joint (i)** & **\(\alpha\) (deg)** & **a (mm)** & **d (mm)** & **\(\theta\) (deg)*** \\ \hline
1 & 0 & 0 & 243.3 & 0 \\
2 & \(-\)90 & 0 & 0 & \(-\)90 \\
3 & 180 & 200 & 0 & \(-\)90 \\
4 & 90 & 87 & 227.6 & 0 \\
5 & 90 & 0 & 0 & 0 \\
6 & \(-\)90 & 0 & 61.5 & 0 \\ \hline \end{tabular} * Joint angle (rotation around the \(z\)-axis).  
\end{table}
Table 2: Modified Denavit–Hartenberg Parameters for Lite 6 [18].  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Joint (i)** & **\(\alpha\) (deg)** & **a (mm)** & **d (mm)** & **\(\theta\) (deg)*** \\ \hline
1 & \(-\)90 & 0 & 243.3 & 0 \\
2 & 180 & 200 & 0 & \(-\)90 \\
3 & 90 & 87 & 0 & \(-\)90 \\
4 & 90 & 0 & 227.6 & 0 \\
5 & \(-\)90 & 0 & 0 & 0 \\
6 & 0 & 0 & 61.5 & 0 \\ \hline \end{tabular}
\end{table}
Table 3: Standard Denavit–Hartenberg Parameters for Lite 6.  
Modified DH Transformation Matrices  
\[T_{i}^{modified}=\begin{bmatrix}\cos(\theta_{i})&-\sin(\theta_{i})&0&a_{i}\\ \sin(\theta_{i})\cos(\alpha_{i})&\cos(\theta_{i})\cos(\alpha_{i})&-\sin(\alpha _{i})&-d_{i}\sin(\alpha_{i})\\ \sin(\theta_{i})\sin(\alpha_{i})&\cos(\theta_{i})\sin(\alpha_{i})&\cos(\alpha_ {i})&d_{i}\cos(\alpha_{i})\\ 0&0&0&1\end{bmatrix} \tag{2}\]  
Combined transformation matrices for both the standard and modified DH parameters, providing a comprehensive description of the end-effector's position and orientation in the Olfactory Lite 6 robot arm.  
Using the modified Denavit-Hartenberg (DH) parameters specific to the Lite 6, we built the transformation matrices for each joint [19].  
\[\textit{Joint 1}:\;T_{1}=\begin{bmatrix}\cos(\theta_{1})&-\sin(\theta_{1})&0&0 \\ \sin(\theta_{1})&\cos(\theta_{1})&0&0\\ 0&0&1&243.3\\ 0&0&0&1\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&243.3\\ 0&0&0&1\end{bmatrix} \tag{3}\]  
\[\textit{Joint 2}:\;T_{2}=\begin{bmatrix}\cos(\theta_{2})&-\sin(\theta_{2})&0&0 \\ 0&0&1&0\\ -\sin(\theta_{2})&-\cos(\theta_{2})&0&0\\ 0&0&0&1\end{bmatrix}=\begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ -1&0&0&0\\ 0&0&0&1\end{bmatrix} \tag{4}\]  
\[\textit{Joint 3}:\;T_{3}=\begin{bmatrix}\cos(\theta_{3})&-\sin(\theta_{3})&0 &200\\ 0&0&1&0\\ -\sin(\theta_{3})&-\cos(\theta_{3})&0&0\\ 0&0&0&1\end{bmatrix}=\begin{bmatrix}0&1&0&200\\ 0&0&1&0\\ -1&0&0&0\\ 0&0&0&1\end{bmatrix} \tag{5}\]  
\[\textit{Joint 4}:\;T_{4}=\begin{bmatrix}\cos(\theta_{4})&-\sin(\theta_{4})&0&8 7\\ 0&0&-1&-227.6\\ -\sin(\theta_{4})&-\cos(\theta_{4})&0&0\\ 0&0&0&1\end{bmatrix}=\begin{bmatrix}1&0&0&87\\ 0&0&-1&-227.6\\ 0&1&0&0\\ 0&0&0&1\end{bmatrix} \tag{6}\]  
\[\textit{Joint 5}:\;T_{5}=\begin{bmatrix}\cos(\theta_{5})&-\sin(\theta_{5})&0 &0\\ 0&0&1&0\\ -\sin(\theta_{5})&-\cos(\theta_{5})&0&0\\ 0&0&0&1\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&0&1&0\\ 0&-1&0&0\\ 0&0&1\end{bmatrix} \tag{7}\]  
\[\textit{Joint 6}:\;T_{6}=\begin{bmatrix}\cos(\theta_{6})&-\sin(\theta_{6})&0 &0\\ 0&0&-1&-61.5\\ -\sin(\theta_{6})&-\cos(\theta_{6})&0&0\\ 0&0&0&1\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&0&-1&-61.5\\ 0&-1&0&0\\ 0&0&0&1\end{bmatrix} \tag{8}\]  
The overall transformation matrix from the base to the end-effector is  
\[T=T_{1}\cdot T_{2}\cdot T_{3}\cdot T_{4}\cdot T_{5}\cdot T_{6} \tag{9}\]  
\[T\;\textit{total}:\;T=\begin{bmatrix}1&0&0&-2.61\\ 0&-1&0&0\\ 0&0&-1&133\\ 0&0&0&1\end{bmatrix} \tag{10}\]  
The translation components are approximately \([-2.61,0,133]\) and the rotation components indicate the orientation in 3D space.