## 4 Simulation and Experiment Validations
Figure 9: Results of fitted curve for the experimental linear-segment toolpath—(**a**) Tool-tip position. (**b**) Tool orientation. The feedrate profiles of the original linear-segment toolpath and the smooth toolpath are compared to show that the total motion duration is reduced by 44.2% from 7.169 s to 3.999 s in Figure 12. It can also be seen that the tool orientation feedrate is generated sharing the same segmented motion time with the tool-tip position. The interpolation is conducted to generate reference positions in the WCS, as shown in Figure 13. From the data, it is evident that the feed process is advanced at high rate except three obtrusive deceleration locations. The tangential kinematic profiles of the tool-tip position and tool orientation are illustrated in Figure 14. It can be spotted that all the tangential kinematic profiles meet with the tangential constraints. In Figure 14c,d, the results of the adjusting techniques mentioned in Section 3.2 is displayed in the feedrate deformation process. Figure 11: Feedrate profile of the tool-tip position with respect to curve parameter. Figure 7:0 Feedrate profile with respect to time—(**a**) Tool-tip position. (**b**) Tool orientation. Figure 7:1 Fitting error of tool-tip position and tool orientation—(**a**) Tool-tip position. (**b**) Tool orientation. The traditional parameter synchronization method obtains the tool orientation feedrate after scheduling the tool-tip feedrate by simply sharing the curve parameter. It may cause wild fluctuation in the tool orientation kinematics, as shown in Figure 7:2  
Since the interpolation results of the tool-tip position and tool orientation are obtained, the reference axial positions are decided according to inverse kinematic transformation from Equation (25). Then the reference axial velocities and accelerations are calculated by differentiating the axial positions. The axial velocities and accelerations does not consider axial constraints are  
Figure 7:3 Tangential kinematic profiles—(**a**) Tool-tip position acceleration. (**b**) Tool-tip position jerk. (**c**) tool orientation acceleration. (**d**) tool orientation jerk. Figure 7:4 Tool orientation kinematic profiles using parameter synchronization—(**a**) Feedrate profile. (**b**) Tangential acceleration profile. Figure 7:5 Fitting error and curvature of planar five-point example. shown in Figure 7:6 It can be seen that despite a slight improvement of feed time (3.595 s), the acceleration of X-axis and velocity of C-axis exceed the corresponding constraints. The axial velocities and accelerations using the parameter synchronization [23] are demonstrated in Figure 7:7 The accelerations of all axes are far greater than the allowable values at certain locations. The axial velocities and accelerations using the proposed method is depicted in Figure 7:8 The data demonstrate that the axial velocities and accelerations are both under limitation. The maximum absolute axial velocities and accelerations using different methods are listed in Table 2 as an intuitive comparison and the anomalously great values are highlighted in bold, especially for axial accelerations using the parameter synchronization method. As can be seen, the large variations of axial accelerations of the traditional parameter synchronization method come from the wild fluctuation in the tool orientation acceleration as shown in Figure 15b. Figure 7:9 Axial velocities and accelerations that does not consider axial kinematic constraints— (**a**) Translational axial velocities. (**b**) Rotary axial velocities. (**c**) Translational axial accelerations. (**d**) Rotary axial accelerations. Figure 10.0 Axial velocities and accelerations using parameter synchronization—(**a**) Translational axial velocities. (**b**) Rotary axial velocities. (**c**) Translational axial accelerations. (**d**) Rotary axial accelerations. The S-shape test piece made of Aluminum Alloy 1060 is machined on the experimental five-axis table-tilting machine tool with a cutter of diameter 10 mm.