## 5 Tool Path Planning Design of Double-Sided Collaborative Machining  
The cutters on both sides are defined as the leading cutter and subsequent cutter, respectively, in the double-sided collaborative machining. In each pair of symmetrical-tool-paths, the cutter that finishes machining first is called the leading cutter, and the other is called the subsequent cutter. The blade surfaces corresponding to the cutters on both sides are called the leading surface and subsequent surface, respectively. In addition, the double-sided collaborative machining also requires six boundary cutter position points to describe the tool path planning process, as shown in Figure 6: speed varied starting point of subsequent surface \(S_{\text{bs}}\), staggered starting point of leading surface \(S_{\text{sl}}\), speed varied ending point of subsequent surface \(E_{\text{bs}}\), speed varied starting point of leading surface \(S_{\text{sl}}\), staggered starting point of subsequent surface \(E_{\text{ss}}\), speed varied ending point of subsequent leading \(E_{\text{sl}}\). Figure 5: The division of the propeller blade surface. \(S_{\text{\tiny{PS}}}\) is the boundary point between the internal region and variable region at the end stage of each symmetrical tool path on the subsequent surface. \(S_{sl}\) is the corresponding point of \(S_{\text{\tiny{PS}}}\) and the starting point of staggered machining. \(E_{\text{\tiny{PS}}}\) is the boundary point between the variable region and edge region at the end stage of each symmetrical tool path on the subsequent surface. \(S_{vl}\) is the boundary point between the edge region and variable region at the start stage of each symmetrical tool path on the leading surface. \(E_{vl}\) is the boundary point between the variable region and internal region at the start stage of each symmetrical tool path on the leading surface. \(E_{\text{\tiny{SS}}}\) is the corresponding point of \(E_{vl}\) and the ending point of staggered machining. The design of the tool path planning process for double-sided collaborative machining through the above six boundary tool points is as follows:  
1. The leading cutter machines the tip region on one side along the machining-tool-path of the leading surface, and the subsequent cutter waits at the zero position;
2. Assuming that the last machining-tool-path of the tip region is path \(n-1\), the leading cutter lifts at \(S_{vl}\) on path \(n\) and waits, while the subsequent cutter starts machining the tip region on one side;
3. When the subsequent cutter machines at the corresponding cutter position point of \(S_{vl}\) on path \(n\), the leading cutter enters and starts double-sided symmetrical machining, as shown in Figure 7a;
4. The subsequent cutter and leading cutter machines to \(S_{\text{\tiny{PS}}}\) and \(S_{sl}\), respectively, on path \(n\), and the subsequent cutter enters the S-type variable speed stage through the sigmoid deformation curve and starts double-sided staggered machining, as shown in Figure 7b;
5. The leading cutter enters the S-type variable speed stage through the sigmoid deformation curve when it completes the machining of path \(n\) and machines to \(S_{vl}\) on path \(n+1\), and the subsequent cutter machines to \(E_{\text{\tiny{PS}}}\) on path \(n\) and exits the S-type variable speed stage at the same time, as shown in Figure 7c;
6. When the subsequent cutter completes the machining of path \(n\) and machines to \(E_{\text{\tiny{SS}}}\) on path \(n+1\), the leading cutter machines to \(E_{vl}\) on path \(n+1\) and exits the S-type variable speed stage, and implement double-sided symmetrical machining again, as shown in Figure 7d;
7. Repeat steps 4 to 6 until the leading cutter and the subsequent cutter return to the zero position. During the single-sided machining of the tip region, the cutter-orientations coincide with the normal direction of the machining cutter position points.