## 2 Improved APF Algorithm
### Path Optimization Based on Steering Model  
Since the APF algorithm essentially calculates the location of the next path point by using the combined force, if the parameters of the force field are not set appropriately, it may lead to discontinuous changes in the path. Meanwhile, if there are multiple obstacles in the environment, cusps or sharp turns in the path can be formed since the formulas for the calculation of the attraction and repulsion forces usually use forms such as the inverse square or the exponential function. Paths are too tortuous, and path cusps occur. It can lead to frequent steering in real-world situations, as well as not meeting the mechanical constraints of the vehicle and leading to very poor comfort. Moreover, the overly zigzag path cannot be used as a reference path for subsequent tracking control studies. Therefore, according to the above problems, this paper adopts the three-point arc method based on the steering model to make the APF algorithm able to make the path more continuous and smooth when performing local obstacle avoidance and path planning. Since the application carrier of the algorithm is a vehicle model, and the vehicle model has the characteristic of a minimum turning radius, in order to comply with the vehicle kinematics constraints, not only are curvature limitations needed on the planned paths, but also the planned paths cannot be less than the minimum turning radius of the vehicle radius, so it is necessary to add the corner constraints in the algorithm. Steering sensitivity is an important index for evaluating the steady-state effect of the vehicle [22], and its formula is shown in Equation (12). \[\frac{\omega}{\delta_{f}}\Bigg{)}_{s}=\frac{\frac{u}{L}}{1+\frac{m}{L^{2}} \times\left(\frac{a}{k_{2}}-\frac{b}{k_{1}}\right)\times u^{2}}\] (12  
After obtaining the Euclidean distances of the three points, the angle formed by the three points is calculated using the cosine theorem, and the angle is restricted to [\(-\)1,1] as shown in Equation (16). \[-1\leq\cos(\theta)=\frac{s01^{2}+s12^{2}-s02^{2}}{2\times s01\times s12}\leq 1 \tag{16}\]  
Based on the angle formed by the three points and the distance between the three points, the radius \(R\) of the three-point arc can be found, and the formula is shown in Equation (17). \[R=\frac{s02}{2\times\sqrt{1-(\cos\theta)^{2}}} \tag{17}\]  
The path planned by the APF algorithm using the theory of the three-point arc method is shown in Figure 13, and the curvature of this smooth path segment is derived and represented in Figure 14.  
As can be seen from Figures 11 and 12, the use of the three-point arc method reduces the path cusp situation while making the path very smooth. However, the path curvature varies greatly, and there are cases of abrupt changes in curvature over a wide range. The curvature reflects the degree of curvature of the path, and the greater the curvature, the sharper the turn of the path. If the curvature of the path is too large, it means that the vehicle needs a smaller turning radius, which is not possible to drive on this path because  
Figure 14: Plot of the rate of change of path curvature after smoothing. Figure 13: Path diagram after smoothing using the three-point arc method. the vehicle model has the limitations of steering angle and minimum turning radius; therefore, this paper proposes a three-point arc method based on the steering model for path optimization. Firstly, according to the vehicle's three-point arc radius \(R\) and vehicle wheelbase \(L\), and ignoring the steering difference between the left and right wheels and mechanical factors. Based on the simplified steering model, to find out the steering angle, the formula is shown in Equation (18). \[direction=\arctan\left(\frac{L}{R}\right)\times\frac{180}{\pi} \tag{18}\]  
where \(L\) is the wheelbase of the vehicle, and \(R\) is the turning radius found by the three-point arc method.