## 3 Results
### Machine Tool Simulation and Manufacturing  
To validate the proposed method, two expectations need to be met. First, the ability to deposit parts on non-planar surfaces. Secondly, to form parts with different dimensions of the contour according to given parameters. Thus, a machine tool was used to form three overhanging hollow blades on a cylindrical hub. Regarding the parameters of the generated paths, they had a layer thickness of 0.4 mm and offset parameters of \(-1.5\) mm, 0.0 mm, and 1.5 mm as shown in Figure 19.  
Before the path data are applied to the actual equipment, they also need to be post-processed according to the positive and negative solution motions [21], not only to set the process parameters but also to be converted into program instructions according to the characteristics of the equipment.  
Figure 17: Test results on the radiation sphere. (**a**) Path contracted inward by 2.0 mm. (**b**) Path expanding outward by 2.0 mm.  
Figure 18: Test results on turbo B. (**a**) Path contracted inward by 2.5 mm. (**b**) Path expanding outward by 2.5 mm.  
In order not to negatively affect the forming stage, a multi-axis machine with an increased rotary table is more suitable for the LMD process with metal powders. Therefore, although the Rotation Tool Center Point (RTCP) subtracts the updating of coordinate data, the tool orientation needs to be decomposed from vectors to rotational angle combinations. The data were converted and confirmed in the machine simulation as shown in Figure 20. They were the coordinates needed for the three moving axes and the angles needed for the two rotating axes.  
The practical application is a coaxial powder-feeding device, which is a five-axis CNC machine equipped with the LMD process module, as shown in Figure 21a. The machine mainly consists of a laser deposition head, a rotary table, and a CNC system. The deposition head is composed of a powder feeder and a laser device that forms a molten pool at a specified location to absorb the metal powder. Figure 21b shows the process of forming a blade with a laser deposition head. The deposition head has only a moving axis and is not capable of rotating, so it is necessary to flip the cylindrical base to avoid collapse and light leakage when forming overhanging structures.  
Regarding the process parameters, the laser power was set at 600 W, the scanning speed was set at 600 mm/min, and the spot diameter was 3 mm. As shown in Figure 22, three blades with different dimensions were formed in different directions of the cylinder. In Figure 22a, the paths were generated with the model boundary offset \(-1.5\) mm. In Figure 22b, the path was generated along the model boundary without offset. In Figure 22c, the path was generated with a model boundary offset of 1.5 mm. Based on the measurements at the same location, it can be seen that the difference between their contour dimensions was around 3 mm, which is consistent with the setting of the offset parameter. The experimental result shows that the paths generated by the proposed method allow the  
Figure 19: Blade paths. Paths with different offset parameters were generated for the same blade to observe the effect of freely forming different contour sizes.  
Figure 20: Machine tool simulation. The path data undergo post-processing to confirm the linkage motion of the deposition head and the rotary table.  
LMD process to form overhanging thin-walled blades on the non-planar surface, as well as to freely offset the paths to form contour boundaries that match the expected dimensions of any machining allowance.  
Figure 21: Coaxial powder-feeding equipment. (**a**) 5-axis CNC machine equipped with the LMD process module. (**b**) The process of forming the blade with a laser deposition head.  
Figure 22: Turbine blades. (**a**) The offset parameter is \(-1.5\) mm. (**b**) The offset parameter is 0.0 mm. (**c**) The offset parameter is \(+1.5\) mm.