## 7 Conclusions  
Robot redundancy has a direct influence on milling. In existing technologies, robot redundancy is generally optimized to improve machining quality and efficiency. In this paper, the ROMJ method and ROMJ-L method are proposed to optimize the robot redundant DOF. The ROMJ method takes the robot's global joint trajectory as the optimization objective, and the joint limit and the tool pose as constraints for optimization. ROMJ-L method is based on the ROMJ method, considering the joint loads of different joints and introducing load coefficients to redesign the global joint trajectory and reduce the rotation of joints with heavy loads. In addition, before optimization, the relationship between chord length and arc length is used to discretize the trajectory points, which greatly reduces the amount of data.  
To prove the effectiveness of the method, the butterfly trajectory is optimized, and the joint trajectory is reduced by 58.8% and 59.6% by the ROMJ method and ROMJ-L method, respectively. In addition, the rotations of joints 1, 2 and 3 with high loads decreased by 4.3% and 9.1% and 58.9%, while the rotations of joints 4, 5 and 6 with low loads increased after the optimization using the ROMJ-L method. Experiments show that the proposed methods can significantly reduce the trajectory of the robot joints, and the introduction of load coefficients can reduce the movement distance of the joints under heavy load, to improve the robot's performance. This method can be applied to milling robots and pave the way for their future market.  
The disadvantage of these methods is that the effect on joint velocities and accelerations is not considered. Therefore, the focus of future research will be on the selection and optimization of joint velocity and acceleration.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Joint Distance (Rad)** & **Joint 1** & **Joint 2** & **Joint 3** & **Joint 4** & **Joint 5** & **Joint 6** \\ \hline Initial & 0.8455 & 1.6155 & 2.1608 & 2.7265 & 5.2734 & 8.4284 \\ ROMJ & 0.4582 & 0.5662 & 1.0169 & 1.0509 & 1.1472 & 1.4683 \\ ROMJ-L & 0.4384 & 0.5149 & 0.4271 & 1.1039 & 1.1634 & 1.4858 \\ \hline The effects of & & & & & & \\ adding \(w_{j}\) & 4.3\% & 9.1\% & 58.9\% & \(-\)5.1\% & \(-\)1.5\% & \(-\)1.2\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: The sum of the kinematic distance of each joint for two optimization objectives.  
Figure 10: Butterfly curve processed by the proposed method.  
**Supplementary Materials:** The following are available online at [[https://www.mdpi.com/article/10.3390/app13095328/s1](https://www.mdpi.com/article/10.3390/app13095328/s1)]([https://www.mdpi.com/article/10.3390/app13095328/s1](https://www.mdpi.com/article/10.3390/app13095328/s1)), Video S1: the processing process by using the Elfin-10 robot.  
Conceptualization, L.L.; Methodology, X.L. and L.L.; Software, X.L.; Validation, X.L.; Investigation, X.L.; Data curation, X.L.; Writing--original draft, X.L.; Writing--review & editing, L.L. and C.F.; Supervision, L.L., C.F., F.L., L.S. and L.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Natural Science Foundation of the Jiangsu Higher Education Institution of China grant number 22 KJB460034 and Undergraduate Training Program for Innovation and Entrepreneurship, Soochow University grant number 202210285051Z.  
**Data Availability Statement:** Not applicable.  
The authors declare no conflict of interest.