## 2 Improved APF Algorithm
### Path Optimization Based on Steering Model
When the APF algorithm plans the location of the predicted points, the coordinates of the three points are used with other parameters of the vehicle model to calculate the arc radius \(R\). If \(R\geq R_{\min}\) also calculates the steering angle based on the steering model, then it means that the planned path angle \(direction\leq direction0\), meets the front wheel angle constraints, and the position of the next point can be planned in accordance with the ensemble of the APF algorithm; If \(R<R_{\min}\) also calculates the steering angle \(direction>direction0\), based on the steering model, that is, if the corner change caused by the resultant force direction is not within the angle constraint range, then the driving direction is re-determined according to Equation (19). \[angle=\pi-2\times\arccos\left(\frac{s01}{2\times R_{\min}}\right) \tag{19}\]  
Equation (18) is based on the geometric characteristics of the vehicle and is used to calculate the front wheel angle of the vehicle for a given turning radius \(R\). The front wheel angle is calculated based on the geometric characteristics of the vehicle. If the calculated front wheel angle does not meet the actual constraints of the vehicle, the limit angle of the vehicle is calculated based on Equation (19), the minimum turning radius of the vehicle, \(R_{min}\), and the Euclidean distance between the first two points. The direction of travel at the limit is shown in Figure 15.  
The three-point arc method using the steering model-based method will be compared with the APF algorithm, and the comparison plot is shown in Figure 16, while the curvature and steering angle of the path will be calculated and plotted as shown in Figures 17 and 18. Figure 15: Direction of travel at the limit. From the figure, we can see that the three-point arc method based on the steering model can well smooth the path planned by APF, and the maximum value of the curvature of the smoothed path is not more than 0.4, and there is no curvature mutation phenomenon, which indicates that the path curvature is not particularly drastic at any point, and it is relatively smooth, and at the same time, the maximum value of the steering angle is 37.7\({}^{\circ}\), and the average value is 7.5\({}^{\circ}\), which is all in the range of the steering angle limitation. \[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]0 Path diagram after smoothing by the three-point arc method based on the steering model. \[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]1 Curvature map of the path after smoothing. \[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]2 Schematic diagram of steering angle. Therefore, the three-point arc method based on the steering model proposed in this paper can largely improve the phenomenon of path cusps produced by APF due to the uneven ensemble force and, at the same time, limit the steering angles to the steering constraints of the vehicle. Since the traditional step size is too small, the step size is optimized according to the density of obstacles in order to improve the efficiency and accuracy of path planning, as well as to ensure smooth paths and avoid large turns. The optimization used in this paper is shown below:  
1. The base step length is set, and a threshold for the number of obstacles is set. When the number of obstacles in a certain range is less than the current threshold, the base step is enlarged, and in this paper, we choose to directly enlarge it 4 times. When the number of obstacles is greater than the threshold, directly shrink 0.8 times. 2. When the number of obstacles is between the maximum and minimum number, a linear difference is calculated for the step size so that the step size can be dynamically adjusted. In this paper, the maximum number of obstacles is set to 6, the minimum number of obstacles is set to 1, and the basic step size in this paper is set to 1. A dynamic step size is used to balance speed and obstacle avoidance.