## 4 Simulation and Experiment Validations  
The feasibility and effectiveness of the proposed method is validated by simulations on a computer with 4.0 GHz CPU and 8 GB RAM and experiments on a five-axis table-tilting machine tool (model of the machine tool) with two rotary axes of A and C whose configuration is illustrated in Figure 7.  
The inverse kinematic transformation of the experimental machine tool is:  
\[\left\{ \begin{array}{l}A=\arccos(o_{k})\\ C=\arctan(o_{i}/o_{j})\\ X=-x\text{cosC}-y\sin C\\ Y=x\cos A\sin C-y\cos A\sin C-z\sin A-L_{AC,z}\sin A\\ Z=x\sin A\sin C-y\sin A\cos C+z\cos A+L_{AC,z}\cos A,\end{array} \right. \tag{25}\]  
where \(L_{AC,z}=40\) mm is the offset distance between A-axis and C-axis in the direction of z-axis. The machine is controlled by a TwinCAT-based CNC system with the sampling period of 1 ms. In order to formulate the feedrate profile to S-shape, we add the maximum allowable feedrate for tool-tip position and tool orientation respectively to the predefined constraints which are listed in Table 1.  
The S-shape test piece is used to analyze the performance of the proposed method. The S-shape test piece is defined by two ruled surfaces and is usually used as a representation of the thin-wall \begin{table}
\begin{tabular}{c c} \hline \hline
**Constraints** & **Values** \\ \hline Interpolation chord error & 1 \(\mu\)m \\ X/Y/Z-axis velocity & 100 mm/s \\ X/Y/Z-axis acceleration & 1000 mm/s\({}^{2}\) \\ A/C-axis velocity & 0.5 rad/s \\ A/C-axis acceleration & 5 rad/s\({}^{2}\) \\ Tool-tip position tangential acceleration & 200 mm/s\({}^{2}\) \\ Tool-tip position tangential jerk & 2000 mm/s\({}^{3}\) \\ Tool orientation tangential acceleration & 5 rad/s\({}^{2}\) \\ Tool orientation tangential jerk & 50 rad/s\({}^{3}\) \\ Tool-tip position maximum feedrate & 50 mm/s \\ Tool orientation maximum feedrate & 0.5 rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Predefined constraints in the experiment. Figure 7: Configuration of experimental five-axis computer numerical control (CNC) machine tool— (a) Real product. (b) Layout. workpiece in the aerospace industry. It requires an allowable range of final error from \(-0.05\) mm to \(+0.05\) mm [35]. We import the CAD model of the test piece shown in Figure 8a to UG NX 10 to plan the toolpath of the ruled surfaces using five-axis flank milling. The programmed toolpath shown in Figure 8b is approximated by a series of discrete G01 data points. The five-axis linear-segment toolpath of one of the four corners which consists of 12 G01 data points shown in Figure 8c is used to demonstrate the effectiveness for the proposed curve-fitting and feedrate scheduling method. The experimental 12 G01 data points is available in Appendix A Table A1. The tool-tip position fitting error tolerance \(\varepsilon_{p}\) is 0.05 mm and the tool orientation fitting error tolerance \(\varepsilon_{o}\) is 0.05\({}^{\circ}\). The results of curve-fitting for the 12-point linear-segment toolpath are shown in Figure 9 and the corresponding fitting error calculated numerically in the curve parameter space is shown in Figure 10. From the data, it is evident that the fitting errors of the tool-tip position and of the tool orientation are both under fitting tolerance. We assign \(N=500\) to compute the discrete locations of the tool-tip position and tool orientation. Then the optimization problem proposed in Section 3.1 is established and solved using the \(linprog\) solver in Matlab to get the allowable feedrate for the tool-tip position. The feedrate profile of the tool-tip position is obtained using the curve splitting and bi-directional scanning techniques and is shown in Figure 11. From the data we can tell that the scheduled feedrate profile is smooth and under feedrate restrictions. Figure 8: Linear-segment toolpath generation for S-shape test piece—(**a**) computer-aided design (CAD) model. (**b**) Toolpath generated by computer-aided manufacturing (CAM). (**c**) Experimental G01 CL data.