## 2 Improved APF Algorithm
### Path Optimization Based on Steering Model
These values are based on many experiments. If the step size is fixed and too large, the algorithm may skip local details, resulting in an unsmooth path, especially in complex environments with many obstacles. Through the adaptive step size, the algorithm can dynamically adjust the step size according to the actual situation to ensure a smoother path and avoid generating large turns or unnatural paths; meanwhile, in the more open area, the gradient change of the potential field is small, and the step size can be increased appropriately to accelerate the search speed, while in the complex environment, the gradient change of the potential field is large, and the step size is reduced to precisely adjust the position to avoid collision or path deviation from the target. Simulation is carried out using MATLAB to add randomly appearing static U-shaped obstacles on the path and two reciprocating dynamic circular obstacles, and the improved algorithm in this paper is compared with the fusion algorithms of A* and DWA, and the comparison graph is shown in Figure 19, and the steering angle is shown in \[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]3  
\[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]4 Algorithm comparison diagrams. They are: (**a**) Obstacle avoidance detail diagram; and (**b**) Algorithm comparison diagram. In the figure, the gray grid is the grid after the expansion process. The purpose is to expand the range of obstacles to provide a safe distance for the global path and does not provide repulsive force. The green curve is the path after the APF planning, and the blue curve is the path after the planning of A* and DWA algorithms. When static U-shaped obstacles and dynamic obstacles suddenly appear on the map, the improved APF algorithm can avoid the local oscillation phenomenon by virtue of the virtual target point to bypass the U-shaped obstacles, and at the same time, it can smoothly avoid dynamic obstacles, and the curvature of the planned path meets the requirements, and the steering angle is within the limitation; while the combination of A* and DWA algorithms is directly trapped in the local optimal phenomenon when facing the U-shaped obstacles. The three-point arc method based on the steering model used in this paper can realize the smoothing of the generated paths in real time during the path planning process, changing the predicted point positions according to random and dynamic obstacles and adjusting the direction angle based on the vehicle model parameters. The real-time smoothed path is compared with the three-times B-spline smoothing, and the comparison graph is shown in \[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]5 The rate of change of curvature of the two will also be compared, as shown in \[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]6  
\[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]7 Comparison chart of three times B spline curve and the smoothing method in this paper. \[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]8 Steering angle diagram. As can be seen from the figure, the path planned by APF is smoothed by using three times B-spline in path post-processing, and its effect is not good at the corners of the path and in avoiding dynamic obstacles, and there are sharp points in the path and the corners are not silky smooth, while the three-point arc method based on the steering model adopted in this paper still has a very good smoothing effect at the corners, and the angle meets the requirements. Meanwhile, the curvature change of the smoothing method used in this paper is better than the three-times B-spline smoothing.