Free-form surfaces have curvature changes in multiple directions and are prone to interference problems. A fixed angle should be maintained between the tool and the surface normal vector at the contact point. Therefore, blade surface machining requires at least five DOF to adjust the position and posture of the tool including three translation freedoms (3T) and two rotational freedoms (2R), as shown in Fig. 7(a). By assigning specific DOF to different actuators, five-axis machine tools can have many configurations. However, as shown in Fig. 7(b), regardless of the configuration, when the tool bypasses the large curvature area, it needs to provide huge concomitant movement.

In this section, the kinematics constraint equations of two five-axis machine tools are established. The influence of tool-paths on machine tool kinematics is quantitatively evaluated using the inverse kinematics solution. When the tool is machining along a path, the drive axis displacement changes smoothly at a lower speed, the machine tool kinematic performance being much improved, and the machining having higher stability and motion control accuracy.

## 3.1 Kinematics constraint equations of two different machine tools

In kinematic analysis, the machine tool can be simplified as a multi-body system that only focuses on the kinematics chain. By establishing a fixed coordinate system on each body, the relative motion between bodies can then be described through coordinate transformation matrixes. The motion constraint equations of different machine tools can be obtained by arranging the coordinate transformation matrixes based on the low-order body array.

Figure 8 shows machine tool I with a serial configuration alongside its structural diagram. The machine tool includes an X-direction guide (1), Y-direction guide (2), polishing tank (3), rotating platform (4), tool rotating platform (6), and Z-direction guide (7). The blade (5) is fixed to the rotating platform (4) and can rotate with it to provide C-direction rotation, \(\tilde {\gamma }\), around the Z-axis. The X-direction guide (1), Y-direction guide (2), and Z-direction guide (7) provide X-direction displacement, \(\tilde {x}\), Y-direction displacement, \(\tilde {y}\),and Z-direction displacement, \(\tilde {z}\), respectively. The tool rotation platform (6) provides B-direction rotation, \(\tilde {\beta }\), about the Y-axis.

The kinematics constraint equation of the machine tool can be expressed as follows:

$$\left[ {\begin{array}{*{20}{c}} { - r} \\ 0 \\ { - R} \\ 1 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\cos (\tilde {\beta })}&0&{ - \sin (\tilde {\beta })}&0 \\ 0&1&0&0 \\ {\sin (\tilde {\beta })}&0&{\cos (\tilde {\beta })}&0 \\ 0&0&0&1 \end{array}} \right] \bullet \left[ {\begin{array}{*{20}{c}} 1&0&0&{ - {x_l}} \\ 0&1&0&{ - {y_l}} \\ 0&0&1&{ - \tilde {z} - {z_l}} \\ 0&0&0&1 \end{array}} \right] \bullet \left[ {\begin{array}{*{20}{c}} 1&0&0&{\tilde {x}} \\ 0&1&0&{\tilde {y}} \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right] \bullet \left[ {\begin{array}{*{20}{c}} {\cos (\tilde {\gamma })}&{ - \sin (\tilde {\gamma })}&0&0 \\ {\sin (\tilde {\gamma })}&{\cos (\tilde {\gamma })}&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right] \bullet \left[ {\begin{array}{*{20}{c}} 0&0&1&0 \\ {-1}&0&0&0 \\ 0&1&0&0 \\ 0&0&0&1 \end{array}} \right] \bullet \left[ {\begin{array}{*{20}{c}} x \\ y \\ z \\ 1 \end{array}} \right]$$

10

where (*x, y, z*) is the position of the tool contact point in the blade coordinate system, (*x**l*, *y**l*, *z**l*) is the position of the origin, *O**t*, of the tool rotation platform coordinate system on the machine coordinate system, *R* is the distance from the end of the tool to *O**t*, and *r* is the tool radius. The axes movement position can be obtained from the inverse kinematics solution of the tool position and posture, as follows:

(11)

where (*P**x*,*P**y*,*P**z*) is the tool vector.

Figure 9 shows machine tool II with a hybrid configuration alongside its structural diagram. The machine tool includes a base (0), Y-direction guide (1), fixed platform (2), 3-RPS parallel motion platform (3), rotating fixture (4), X-direction guide (5), abrasive belt tool (6), and clamper (7). The clamper (7) fixes the blade to the rotating fixture (4), which provides B-direction rotation, \(\tilde {\beta }\), around the Y-axis. The X-direction guide (5) and Y-direction guide (1) provide X-direction displacement, \(\tilde {x}\), and Y-direction displacement, \(\tilde {y}\), respectively. The 3-RPS parallel motion platform (3) locks the rotation freedom around the Y-axis, only providing Z-direction displacement, \(\tilde {z}\), and A-direction rotation around the X axis.

The kinematics constraint equation can be expressed as follows:

$$\left[ {\begin{array}{*{20}{c}} {{x_t}} \\ {{y_t}} \\ {{z_t}} \\ 1 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 1&0&0&{ - \tilde {x}} \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right] \bullet \left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&{\tilde {y}} \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right] \bullet \left[ {\left[ {\begin{array}{*{20}{c}} 1&0&0&{{x_p}} \\ 0&{\cos (\tilde {\alpha })}&{ - \sin (\tilde {\alpha })}&{r\cdot (1 - \cos (\tilde {\alpha }))+{y_p}} \\ 0&{\sin (\tilde {\alpha })}&{\cos (\tilde {\alpha })}&{ - {z_p} - \tilde {z}} \\ 0&0&0&1 \end{array}} \right]} \right]\left[ {\begin{array}{*{20}{c}} {\cos (\tilde {\beta })}&0&{\sin (\tilde {\beta })}&{{x_r}} \\ 0&1&0&{{y_r}} \\ { - \sin (\tilde {\beta })}&0&{\cos (\tilde {\beta })}&{{z_r}} \\ 0&0&0&1 \end{array}} \right] \bullet \left[ {\begin{array}{*{20}{c}} x \\ y \\ z \\ 1 \end{array}} \right]$$

12

where (*x, y, z*) is the position of the tool contact point in the blade coordinate system, (*x**r*, *y**r*, *z**r*) is the position of the origin, *O**b*, of the blade coordinate system on the motion platform coordinate system, (*x**p*, *y**p*, *z**p*) is the position of the origin, *O**m*, of the motion platform coordinate system on the fixed platform coordinate system in the initial state, and (*x**t*, *y**t*, *z**t*) is the position of the tool nose on the machine coordinate system in the initial state. \(\tilde {z}\) and \(\tilde {\alpha }\) are coupled by the linear motion of three parallel drive axes, which can be calculated as follows:

$$\begin{gathered} {L_1}{\text{=}}\sqrt {{{(\frac{3}{2} \bullet {r_b} \bullet \cos (\tilde {\alpha }) - \frac{1}{2} \bullet {r_b} - {r_p})}^2}+{{( - {r_b} \bullet \sin (\tilde {\alpha })+z+\tilde {z})}^2}} \hfill \\ {L_2}{\text{=}}{L_{\text{3}}}{\text{=}}\sqrt {{{({r_b} - {r_p})}^2}+{{(\frac{1}{2} \bullet {r_b} \bullet \sin (\tilde {\alpha })+z+\tilde {z})}^2}} \hfill \\ \end{gathered}$$

13

where \({r_b}\) and \({r_p}\) are the distances from the center of the motion platform and the fixed platform to the hinge point, respectively.

## 3.2 Kinematics analysis of the tool-paths

Based on the inverse kinematics solution, the machine tool drive axes displacement of traditional paths is as shown in Fig. 10. Although the configurations differ, the two machine tools show similar kinematic characteristics. In the large curvature area where the surface normal vector changes rapidly, the kinematic characteristics of the machine tool deteriorate. The displacement of multiple drive axes related to the tool posture adjustment changes rapidly over a small range, which is differs significantly from the displacement of a flat area. When machining this area, the drive axes need not only high-speed motion but also rapid acceleration. Multi-axis dynamic coordination is required to ensure the accuracy of tool movement, which increases the difficulty of machine tool control and the probability of machining defects. The transverse and pocket paths extend along the direction of the blade, and unavoidable tool posture adjustment is required. However, two longitudinal paths extending along the length direction still cannot avoid this problem in key areas, as the paths cannot adapt to the surface curvature.

In the proposed method, this problem is completely solved. Fig. 11 shows the machine tool drive axes displacement of the proposed path. In the trailing-edge area—with its complex curvature—the axis displacement still changes smoothly without any rapid variations. Compared with the traditional paths near the trailing-edge area, the axis rotation is reduced by 90%, and the axis motion by 60%.

When the feed rate is constant at 500 mm/min, the axis motion speed is as shown in Fig. 12. The kinematic characteristics of the pocket path are similar to those of the transverse path, so they need no longer be compared. Among the three comparison methods, the longitudinal iso-parametric path is the least affected by the tool posture adjustment. The maximum rotation and motion speeds of the proposed path are only 1.2% and 10% of the longitudinal iso-parametric path, respectively. Concomitant movement in the LTE area is reduced by 98%. The speed of the axis is much lower than comparison methods, which improves the kinematic characteristics of the machining.

Suppose the speed limit of the linear axis is 100 mm/s, and the speed limit of the rotary axis is 30°/s. The maximum speed that the tool can reach under the axis limit is as shown in Fig. 13. In the traditional paths, the tool feed speed is restricted in the LTE area, reducing the machining efficiency and causing uneven removal. In the proposed path, the feed speed limit is always close to the speed limit of the linear axis and is not restricted by the concomitant motion axis, so it has excellent kinematic characteristics. The minimum feed speed is four times that of the traditional longitudinal iso-parametric path.