## 6 Conclusions  
This paper investigates the analytical model of the cutter-workpiece engagement angle and the prediction method of the cutting forces in the ellipse-based trochoidal milling process. Our conclusions are summarized as follows:  
1. An ellipse-based trochoidal toolpath model is developed by extending the true trochoidal toolpath model with a control parameter, namely the compression ratio in the trochoidal step direction.
2. The analytical milling contour equation corresponding to the ellipse-based trochoidal toolpath is presented and the effective revolution interval within the actual cutting process is determined.
3. An analytical calculation method for the start and exit angles for the ellipse-based trochoidal milling process is proposed.
4. The compression ratio has significant effects on the trochoidal toolpath length and the cutter-workpiece engagement angle. The smaller the compression ratio, the shorter the overall trochoidal toolpath length, and the higher the material removal rate. As the compression ratio decreases, the engagement angle first becomes more balanced within the effective milling interval and then increases steeply in the initial and final segments. In this research, a compression ratio of 50% is optimal.
5. The effect of the cutter radius on the cutter-workpiece engagement angle is minimal. A larger cutter radius does not cause an obvious change in the engagement angle. However, a larger cutter radius results in a smaller semi-major axis of the fundamental ellipse, which helps to shorten the total trochoidal toolpath length.
6. The cutting-force coefficients for the milling of titanium alloy Ti-6 Al-4V with a flat-end cutter are identified. Several ellipse-based trochoidal milling experiments with different toolpath parameters are conducted to verify the accuracy and reliability of  
Figure 17: Predicted total cutting force during trochoidal milling for different toolpath parameters. \(a=15\) mm, \(R_{c}=5\) mm. (**a**) \(b=15\) mm, \(\rho=100\%\), \(s=0.6\) mm; (**b**) \(b=11.25\) mm, \(\rho=75\%\), \(s=0.6\) mm; (**c**) \(b=7.5\) mm, \(\rho=50\%\), \(s=0.6\) mm; (**d**) \(b=3.75\) mm, \(\rho=25\%\), \(s=0.6\) mm.  
the proposed method. It can be found that the predicted results agree well with the experimental data.
* The trends in cutting force are consistent with the trends in cutter-workpiece engagement angle for different trochoidal toolpath parameters.  
Conceptualization, X.Z. and D.Z.; methodology, X.Z., J.Z. and Q.Q.; software, X.Z. and J.Z.; validation, Q.Q. and J.Z.; formal analysis, C.Z. and D.Z.; data curation, X.Z. and Q.Q.; writing--original draft preparation, X.Z.; writing--review and editing, C.Z. and J.Z.; funding acquisition, X.Z., J.Z. and C.Z. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Key Research and Development Program of China, grant number 2020 YFF0304905; the National Natural Science Foundation of China, grant number 52075451; the Science Center for Gas Turbine Project, grant number P2022-B-IV-012-001; and the Aeronautical Science Foundation of China, grant number 2019 ZE053008.  
Not applicable.  
The authors declare no conflict of interest.