## 3. Cutter Position Points Interpolation and Data Alignment on Both Sides  
The tool path curves between adjacent tool position points are fitted by piecewise polynomial interpolation during machining, but unrelated fluctuations to the ideal curve may occur in a series of cutter position points fitting processes. The causes of unrelated fluctuations are generally uneven cutter position points or excessive spacing, which can also lead to discontinuity in the cutter-orientation transformation. Therefore, in this paper, a cubic spline curve is applied to fit the cutter position points path and cutter tail points path and perform equidistant interpolation on the fitting curve after determining the number of interpolation steps, as shown in Figure 4. In the machine coordinate system, the cutter-orientation \((i_{k,n},j_{k,n},k_{k,n})\) of the interpolation point is obtained by using the interpolation cutter position point coordinates \((x_{k,n},y_{k,n},z_{k,n})\) and its corresponding interpolation cutter tail point coordinates \((x_{k,n},y_{k,n},z_{k,n})\). Figure 3. Symmetrical XYZ-3 RPS double-sided hybrid machine tool: (1) machine base; (2) X-axis lead screw; (3) workbench; (4) propeller blade workpiece; (5) clamping support; (6) 3 RPS parallel mechanism; (7) Y-axis lead screw; (8) Z-axis lead screw; and (9) ball nose end mill. Figure 2. Symmetrical-cutter-position-points of symmetrical-tool-paths. The number of original cutter position points on both sides is inconsistent due to the complex shapes of the blade curved surfaces, and it is impossible to achieve double-sided symmetrical machining. Therefore, it is necessary to achieve the correspondence of cutter position points on both sides through data alignment processing. According to the number of original cutter position points on both sides of the blade, select the side with fewer cutter position points as the projection surface and the other side as the template surface. Firstly, interpolate the cutter position points on the template surface. Each tool path between two original cutter position points is regarded as an interpolation interval, and it is assumed that there are \(K\) interpolation intervals in a machining-tool-path. In the machine coordinate system, the interpolation interval starting point coordinates \(p_{k}=\left(x_{k},y_{k},z_{k}\right)^{T}\) and orientations \(w_{k}=\left(i_{k},j_{k},k_{k}\right)^{T}\), and the ending point coordinates \(p_{k+1}=\left(x_{k+1},y_{k+1},z_{k+1}\right)^{T}\) and orientations \(w_{k+1}=\left(i_{k+1},j_{k+1},k_{k+1}\right)^{T}\), where \(k=1,2,\cdots,K\). Thereby, the starting points and ending points of the cutter tail point interpolation intervals are obtained as follows:  
\[{p_{k}}^{\prime}= \begin{bmatrix}{x_{k}}^{\prime}\\ {y_{k}}^{\prime}\\ z_{k^{\prime}}\end{bmatrix} = \begin{bmatrix}{x_{k}}\\ {y_{k}}\\ {z_{k}}\end{bmatrix} +d_{T} \begin{bmatrix}i_{k}\\ j_{k}\\ k_{k}\end{bmatrix} , \tag{1}\]  
\[{p_{k+1}}^{\prime}= \begin{bmatrix}{x_{k+1}}^{\prime}\\ {y_{k+1}}^{\prime}\\ {z_{k+1}}^{\prime}\end{bmatrix} = \begin{bmatrix}{x_{k+1}}\\ {y_{k+1}}\\ {z_{k+1}}\end{bmatrix} +d_{T} \begin{bmatrix}i_{k+1}\\ j_{k+1}\\ k_{k+1}\end{bmatrix} , \tag{2}\]  
where \(d_{T}\) represents the tool length. A single machining-tool-path is a spatial curve, and the coordinates of the Y-axis monotonically increase or decrease in the machine coordinate system.