# Analysis and Simulation of Polishing Robot Operation Trajectory Planning
## 2 Materials and Methods  
### The Establishment of the Robotic Arm Kinematics Model and the Spatial Spiral Curve Model  
#### 2.1.1 Robot Kinematics and Modeling  
The ABB-IRB120, as a compact six-axis industrial robot, has been widely adopted in the manufacturing industry due to its high precision, exceptional flexibility, and reliability. Figure 1 illustrates the planar schematic diagram of the ABB-IRB120 six-axis robot, which provides a clear visual representation of the robot's structural composition and relationship between its various links. This schematic serves as a foundational basis for understanding the robot's kinematic and dynamic characteristics. The link parameters of the ABB IRB120 robot are critical in defining its motion capabilities and performance. The link parameters for the ABB IRB120 polishing robot are referenced in the literature [8,21]. These parameters, such as link lengths, twist angles, and offset distances, are essential components of the robot's kinematic model and are used to calculate its forward and inverse kinematics. Table 1 presents the comprehensive DH (Denavit-Hartenberg) parameter table for the ABB-IRB120 six-axis robot. By using these parameters, it is possible to accurately determine the position and orientation of the robot's end-effector relative to its base frame. \begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline
**Link Rod \(i\)** & **Twist Angle \(\boldsymbol{\alpha_{i-1}}\)/Rad** & **Link Length \(\boldsymbol{a_{i-1}}\)/mm** & **Joint Angle \(\boldsymbol{\theta_{i}}\)/rad** & **Link Offset \(\boldsymbol{d_{i}}\)/mm** & **Other** \\ \hline
1 & 0 & 0 & \(\theta_{1}\) & \(\mathrm{l_{1}}\) & \(\mathrm{l_{1}}=290\) mm \\
2 & \(-\pi/2\) & 0 & \(\theta_{2}-\pi/2\) & 0 & — \\
3 & 0 & \(\mathrm{l_{2}}\) & \(\theta_{3}\) & 0 & \(\mathrm{l_{2}}=270\) mm \\
4 & \(-\pi/2\) & \(\mathrm{l_{3}}\) & \(\theta_{4}\) & \(\mathrm{l_{4}}\) & \(\mathrm{l_{3}}=70\) mm \\
5 & \(\pi/2\) & 0 & \(\theta_{5}+\pi\) & 0 & — \\
6 & \(\pi/2\) & 0 & \(\theta_{6}\) & \(\mathrm{l_{5}}\) & \(\mathrm{l_{5}}=72\) mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The D-H parameters table for the ABB-IRB120 six-axis robot. Figure 1: Planar schematic of the ABB-IRB120 six-axis. In this setup, the coordinate system for the robotic arm's links is established using the modified DH parameters method. When studying a generalized linkage system with \(n\) joints, the homogeneous transformation matrix between adjacent links is considered, considering the neighboring links and joint \(\text{Axis}_{i-1}\), \(\text{Axis}_{i}\), and \(\text{Axis}_{i+1}\), as shown in Figure 2.  
The transformation matrix for the linkage of the improved DH coordinate system is as follows [10]:  
\[{}^{i-1}T_{i}= \begin{vmatrix}\cos\theta_{i}&-\text{sin}\theta_{i}&0&a_{i-1}\\ \cos a_{i-1}\text{sin}\theta_{i}&\cos a_{i-1}\text{cos}\theta_{i}&-\text{sin}a _{i-1}&-d_{i}\text{sin}a_{i-1}\\ \sin\pi_{i-1}\text{sin}\theta_{i}&\sin\pi_{i-1}\text{cos}\theta_{i}&\cos a_{i- 1}&d_{i}\text{cos}a_{i-1}\\ 0&0&0&1\end{vmatrix}  \tag{1}\]  
where \(\alpha_{i}\), \(a_{i}\), and \(\theta_{i}\) are the linkage parameters. Then, the positive kinematic equation of the robot is obtained from the pose transformation matrix of each joint of the robot [22]:  
\[{}^{0}T_{6} ={}^{0}T_{1}(\theta_{1})\cdot\ ^{1}T_{2}(\theta_{2})\cdot\ ^{2}T_{3}(\theta_{3})\cdot\ ^{3}T_{4}(\theta_{4})\cdot\ ^{4}T_{5}(\theta_{5})\cdot\ ^{5}T_{6}( \theta_{6}) \tag{2}\] \[= \begin{bmatrix}n_{x}&\alpha_{x}&x&p_{x}\\ n_{y}&\ o_{y}&a_{y}&p_{y}\\ n_{z}&\ o_{z}&a_{z}&p_{z}\\ 0&0&0&1\end{bmatrix}