## 5 Conclusions  
This paper proposes a double B-spline curve-fitting and synchronization-integrated feedrate scheduling method for five-axis linear-segment toolpath. The main technical characteristics are as follows--(1) the tool-tip position and tool orientation are fitted by two cubic B-spline curves respectively so that the discontinuities caused by transition corners of five-axis linear-segment toolpath are handled and the fitting error is strictly respected; (2) the axial kinematic constraints are mapped and satisfied in the feedrate scheduling process of the tool-tip position; (3) the synchronization and smooth rotary motion is obtained using feedrate deformation for tool orientation. Simulations and experiments are carried out to validate the feasibility and effectiveness of the proposed method. For a toolpath of 12 CL data points from the S-shape test piece, the proposed curve-fitting method constrains the fitting error under fitting tolerance. The proposed machining time is reduced significantly by 44.2% in comparison with the original linear-segment toolpath. To verify  
Figure 20: Real axial kinematics—(**a**) Translational axial velocities. (**b**) Rotary axial velocities. (**c**) Translational axial accelerations. (**d**) Rotary axial accelerations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{2}{c}{**Position Error (mm)**} & \multicolumn{2}{c}{**Orientation Error (deg)**} \\ \cline{2-5}  & **Max.** & **Average** & **Max.** & **Average** \\ \hline Parameter synchronization & 0.0525 & 0.0268 & 0.0717 & 0.0378 \\ Proposed & 0.0507 & 0.0243 & 0.0512 & 0.0317 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitting error of the experiments. the smooth motion of the proposed feedrate scheduling method, we compare the proposed method with the method does not consider axial kinematic constraints and the method using parameter synchronization. The results show that the proposed rotary motion is smooth and the axial motions are under kinematic limits. The experiments are conducted on the S-shape test piece. The real tool-tip position and tool orientation are reversely computed using the real axial positions to confirm that the proposed method can generate a smooth toolpath with allowable fitting error. The real axial kinematics are further smoothed in the real machining process. Despite the progress made in this study, only fitting error and smoothness are considered for the fitted curve. However, the machining accuracy is affected by other factors such as thermal error, geometric errors of machine tool, dynamic error and so forth, and workpiece measurement to evaluate the surface quality directly is not conducted. Therefore, further research should be done in our future study. Conceptualization, X.G. ; Formal analysis, S.Z., L.Q., X.L., Z.W. and Y.W. ; Investigation, S.Z., L.Q., X.L., Z.W. and Y.W. ; Methodology, X.G. ; Project administration, S.Z. and L.Q. ; Software, X.G. ; Writing--original draft, X.G. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China grant number 51675478 and 51875516, and the Natural Science Foundation of Zhejiang Province grant number LY18E050001. The authors declare no conflict of interest. * [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON]; [PERSON] A G4 continuous B-spline transition algorithm for CNC machining with jerk-smooth feedrate scheduling along linear segments. _Comput. Aided Des._**2019**, _115_, 231-243. [CrossRef]
* [PERSON] et al. (2015) [PERSON]; [PERSON]; [PERSON] A realtime curvature-smooth interpolation scheme and motion planning for CNC machining of short line segments. _Int. J. Mach. Tools Manuf._**2015**, _96_, 27-46. [CrossRef]
* [PERSON] et al. (2016) [PERSON]; [PERSON]; [PERSON]; [PERSON] Real-time Bezier interpolation satisfying chord error constraint for CNC tool path. _Sci. China Technol. Sci._**2016**, _59_, 203-213. [CrossRef]
* [PERSON] et al.