In recent years, the machine tool manufacturing industry has developed rapidly, and Computer Numerical Control machine tools (CNC-MT) have covered all manufacturing industries. Digital control utilizes digital technology to manage the movement and cutting of machine tools, including mechanical manufacturing, inspection, automatic control, and computer technology, with advantages such as high machining accuracy and fast machining efficiency. With the rapid development of aerospace, national defense, new energy and other fields, industries with complex structures are rapidly rising (Kiswanto, 2020; Liu et al., 2021; Guo et al., 2023). Integrated turbine blades (ITB), as complex thin-walled components, are widely used in fields such as water conservancy and aviation, and their proportion in aviation generators reaches 30%. Against the backdrop of increasing demand in the national defense market, the production requirements of ITB are developing towards high precision and efficiency. Different from open and semi open, due to the thinner ITB blades and narrower distance between blades, there is not enough machining space for the cutting tools during the blade machining process. At the same time, due to the existence of non expandable ruled surfaces (NERS) with fundamental errors, it is difficult for the tool edge to fully fit with the blade surface, resulting in insufficient or excessive cutting between the NERS and the tool envelope surfaces (TES), which greatly increases the difficulty of ITB machining. The methods of machining complex surfaces can be divided into point milling and side milling. The former is accurate in machining, but the tool path range is small, so the consistency of surface machining of parts is low (Cheng et al., 2020; Franciosa et al., 2020; Wei Y. et al., 2021). Therefore, this study will be based on Non Uniform Rational B-Spline (NURBS) curves, with the goal of reducing over cutting or under cutting on the surface of ITB blades, and conduct in-depth analysis of NERS path planning through side milling. In addition, the path planning model is solved by integrating Simulated Annealing (SA) and Particle Swarm Optimization (PSO) algorithms to iteratively find the optimal path. The innovation of the research can be mainly summarized as the following two points. One is to construct NER based on NURBS and analyze the principle error of NER in the motion trajectory of ITB tool axis. The second is to use the SA-PSO fusion algorithm to solve the path planning model of NERS for side milling in this study. This study will elaborate on it from the following four aspects. Part 1 elaborates on the current research status of CNC-MT technology and ITB side milling at home and abroad. Part 2 introduces the NERS path planning for ITB side milling, with a focus on the SA-PSO fusion algorithm. Part 3 analyzed the effectiveness and feasibility of the NERS path planning for ITB side milling. Part 4 summarizes the research results and proposes the development direction for the next step of research.

CNC technology has been widely applied in various manufacturing fields such as aerospace. At the same time, some achievements have been made in the research of blade path planning and impeller models. Wei and other researchers proposed trajectory planning for conical cutter side milling of integral impellers, obtaining the cutting and expanding tool trajectories through NURBS curves. At the same time, the five axis side milling method was used for precision machining of the blades of the integral turbine. The virtual simulation system of the machining center confirmed that compared to the ball end milling cutter, the tool path trajectory was finer, the machining accuracy was higher, and it was easier to optimize the tool path. The machining accuracy error was 0.007–0.012 mm (Wei J. et al., 2021). Yu et al. proposed a new calculation method for the error of the R-offset tool axis for the tool axis trajectory planning problem of NERS machining with conical cutting tools, and derived an accurate analysis using a formula. This method could obtain the theoretical machining error of each point on the straight generatrix, and the test results further verified this conclusion. The difference between the machining error and the theoretical error was almost 0 (Yu et al., 2021). Feng et al. proposed a U-shaped milling method that combines traditional cycloidal milling and side milling. This method defined the feature parameters of the path and used the concatenated micro arc mapping algorithm to map the three-dimensional boundaries to obtain the distribution pattern of tool contact points. The three-axis milling groove test showed that the peak and average cutting forces decreased by 25% and 60%, respectively. Therefore, this method could reduce the instability of cutting force, improve machining efficiency, and reduce the degree of tool wear (Feng et al., 2021). Zhao et al. proposed a structure that utilizes NiAl based high-temperature alloy for milling precision of narrow blade turbine impellers to achieve the milling accuracy of typical thin-walled structural components of turbine engines. It determined optimization parameters through milling force and tool wear behavior, and proposed a flexible iterative compensation method to reduce machining errors in impeller milling. The milling accuracy of turbine impeller had significantly improved (Zhao et al., 2020). Fan et al. proposed a five axis milling method based on NURBS curves and applied it to centrifugal impellers with crowns. They also analyzed the calculation method of the final section. The machining accuracy of the crown centrifugal impeller was significantly reduced, and this method had certain feasibility in the machining field (Fan et al., 2020).

Yan’s team analyzed the path planning problem of conical tool side milling method and proposed a solution algorithm using immune PSO and least squares method. The objective function was set as the error measurement function for each tool position, and the initial tool position was determined by the two-point offset method. The machining error of this method was reduced by 86% relative to the specified error (Yan et al., 2021). Serin et al. conducted in-depth analysis on integrated energy-saving machining of rotating impellers and optimized the objective function using PSO intelligent algorithm. This scheme could achieve precision machining of turbine blades (Serin et al., 2020). Researchers such as Tüchler used a hybrid algorithm of PSO and genetic algorithm to optimize the rotor shape profile, wall thickness, and number of channels of a wave rotor. This confirmed that the numerical optimization method has more advantages compared to traditional methods (Tüchler and Copeland, 2021). Guo and other scholars analyzed the optimization model of surface roughness and cutting force using multi-objective PSO to achieve the accuracy and speed of nickel based high-temperature alloy processing. The relative error between the true value of tangential cutting force and the predicted value of the model was 8.14% (Guo et al., 2022).

Based on the current research status of integrated impeller CNC machining both domestically and internationally, in-depth research has been conducted on the design and accuracy improvement of various tool paths. The commonly used four point bias method and particle swarm optimization algorithm have better matching performance compared to other algorithms. However, the four point equidistant offset method may suffer damage from the previous tool’s machining position and cannot achieve the expected machining effect. Therefore, it cannot be used for actual CNC-MT machining. This study will take the side milling machining method as an example and solve the path planning model of the side milling cutter axis using the SA-PSO fusion algorithm, in order to achieve the maximum approximation of TES to the design surface.

High speed machining technology has evolved from three-axis machining to five axis machining. Five axis machining is currently an important means of machining complex surfaces. The CNC machining machine tool for turbine blades differs from ordinary machine tools in that it has high precision and manufacturing efficiency. And it has an impeller digital processing module, which can complete the processing of different types of surfaces such as ruled surfaces, NER surfaces, and complex surfaces. There are two problems in current research, namely, the design of surfaces and TES deviations for single tool path planning, as well as the high computational complexity and low computational efficiency of the constructed mathematical model. This study will take the side milling machining method as an example and solve the side milling cutter axis path planning model using the SA-PSO fusion algorithm. The key to this model is to find the machining pose at each moment, so that the envelope surface of the tool is consistent with the design surface.

The original data of the blades in this study was obtained from a certain enterprise’s integral impeller. A single blade was divided into 10 blade surfaces, and the third blade surface was selected as an example. The parameters include the wires and busbars of the curved surface. Select 8 tool positions uniformly on the second curved surface, and select eight points uniformly on the axis in the direction of the generatrix as the judgment positions for error measurement. Simulate using a column cutter with a diameter of 6 mm. Figures 4A,B show the control points for the top and bottom wires in NURBS, respectively.

Figures 5A,B show the training curves for single blade position SA and SA-PSO, respectively. In Figure 5A, it can be seen that under the control of the PSO algorithm, after 25 rounds of training for a single tool position, it repeatedly falls into the local optimal solution, with stable error values of 1.210, 1.302, and 1.345, respectively. Figure 5B shows that after adjusting the error using the SA-PSO algorithm, a single tool position avoids the phenomenon of getting stuck in local optimal solutions multiple times. The final value range is −0.416 to 0.421. In the subsequent training process, it is necessary to increase the maximum step distance of a single step and reduce the cutting amount of the tool under some training times. The result of a single tool path has certain significance for the final ITB CNC machining technology, as it can directly determine the machining quality of ITB and the operating efficiency of the tool.

Figures 6A,B show the optimization results and optimization time of PSO and SA-PSO algorithms at different iteration times, respectively. As the number of iterations increases, the error values of both PSO and SA-PSO algorithms gradually decrease, with convergence times of approximately 7 and 6, respectively. The stable error value of SA-PSO algorithm is 0.253, which is 30.45% lower than PSO algorithm. For optimization time, the stable values of PSO and SA-PSO algorithms are 3.2 and 3.3 s, respectively. This indicates that although the iteration process of SA-PSO algorithm is not as fast as that of a single PSO algorithm, it is still within a reasonable numerical range and the numerical difference is not significant. Therefore, due to the significant advantage of the SA-PSO algorithm proposed by the research institute in terms of error value and the smaller difference in running time compared to the PSO algorithm, the overall performance is better.

Figure 7 shows the envelope error diagram of the tool at its initial position When the tool is in its initial position, the range of envelope error is −0.5 to 0 mm. This indicates that there is a situation of over cutting between the design surface and the tool. Therefore, the design surface needs to be adjusted through numerical algorithms to match the actual motion trajectory.

Figure 8 shows the impact of different algorithms on the calculation results of envelope errors. Figures 8A,B show the single tool position envelope error diagrams for PSO and SA-PSO algorithms, respectively. In Figure 8A, the range of envelope error values is −0.018 to 0.018 mm. The results of the single tool position envelope error graphs for both SA-PSO algorithms show that the range of envelope error values is −0.01 to 0.01 mm. Therefore, after optimizing the trajectory model of the side milling cutter axis using two algorithms, the relationship between the design surface and the tool is still distributed between the over cut and undercut states. However, the SA-PSO fusion algorithm has significantly reduced the envelope error after optimization.

Table 2 shows the envelope error of the single tool path for PSO and SA-PSO algorithms. When the calculation reaches the 7th iteration, both optimization algorithms have reached the optimal solution. The range values of PSO and SA-PSO algorithms are 0.0213 and 0.0165 mm, respectively. The range mean of the SA-PSO fusion algorithm is reduced by about 15% compared to a single optimization algorithm.

Figures 9A,B show the top and bottom positions of the initial and SA-PSO optimized tool axis trajectory surfaces, respectively. The tool axis trajectory surface optimized by SA-PSO is closer to the actual trajectory, and the processed blades better meet the quality requirements of ITB. The initial and optimized tool axis trajectory surfaces exhibit significant errors in the Y-axis direction, which may be caused by CNC-MT. After optimizing the PSO algorithm through the SA algorithm, this method can effectively solve the problem of difficulty in obtaining global optima due to the rapid convergence of population individuals, and is easy to overcome the advantage of local minima.

Figures 10A,B show the training results of using wire overlap as the target in both single and global cutting scenarios, respectively. Overall, after each training period, the sum of the normal and position mean squared deviations of the target model and the initial blank shows a large fluctuation range. The range of the sum of mean squared deviations for single and global cutting is 0.0011–0.0198 and 0.046–0.0341, but the overall error value is relatively small. This indicates that only some position target models match the normal direction, but the solution algorithm of the designed side milling cutter axis trajectory planning model can achieve good iterative operation results.

To further validate the application effect of SA-PSO algorithm in lateral milling cutter axis trajectory planning, this study compared it with other similar studies, including references (Lu et al., 2022; Zhao et al., 2023; Zou et al., 2022). A comparison of the trajectory planning effects of the integral impeller side milling cutter shaft was conducted on a five axis machine tool model HS5AXIS 350, with indicators including average error and over cutting rate. The results are shown in Table 3. It can be clearly seen from Table 3 that the over cutting rate and machining error of the SA-PSO algorithm are 46.32% and 0.142 mm, respectively, and the required time is only 4.512 s. Compared with other studies, the SA-PSO algorithm has better efficiency and effectiveness in planning the axis trajectory of side milling cutters.

The problem of deviation design between the design surface and TES when planning a single tool position. To solve the mathematical model through reasonable numerical methods to reduce computational complexity and improve operational efficiency, this study proposed to use the SA-PSO optimization algorithm to solve the trajectory planning model of the side milling cutter axis. After 25 rounds of training with a single blade position, it repeatedly fell into the local optimal solution, with stable error values of 1.210, 1.302, and 1.345, respectively. After adjusting the error through the SA algorithm, a single tool position avoided the phenomenon of getting stuck in local optimal solutions multiple times. The final value range was −0.416 to 0.421. For optimization time, the stable values of PSO and SA-PSO algorithms were 3.2 and 3.3 s, respectively. When the tool was in its initial position, the range of envelope error was −0.5 to 0 mm, and there was a situation of over cutting between the design surface and the tool. The range of single tool position envelope errors for PSO and SA-PSO algorithms was −0.018 to 0.018 mm and −0.01 to 0.01 mm, respectively. When the calculation reached the 7th iteration, both PSO and SA-PSO optimization algorithms had reached the optimal solution, with range values of 0.0213 and 0.0165 mm, respectively. The range mean of the SA-PSO fusion algorithm was reduced by about 15% compared to a single optimization algorithm. The tool axis trajectory surface optimized by SA-PSO was closer to the actual trajectory, and the processed blades better met the quality requirements of ITB. The target model at some positions matched the normal direction, but the solution algorithm of the designed lateral milling cutter axis trajectory planning model could achieve good operational results and find the optimal path with the minimum error in a fixed area. This study has shown good results in the field of principle error resolution, but there are still shortcomings. In the subsequent ITB CNC operation process, it is necessary to consider the impact of machine tool driven tool vibration and impeller defects on the actual effect.