## 4 Improved Artificial Potential Field Method  
### Improved Attraction Field  
A large number of scholars have improved and optimized the traditional artificial potential field method from various aspects. Most of them paid attention to modifying the potential field function, and there are some reasons for this, as follows: (1) the expression of the potential field function is easy to modify; (2) the potential field is more in line with actual physics, so it is easy to understand; (3) the modification of the potential field can improve the efficiency of the algorithm in essence. Considering the following reasons, this paper focuses on the improvement of the attraction potential field.  
First of all, the core of the local minimum problem is that the values of the attraction force and repulsion force are the same, and the directions are opposite, which can be improved by modifying one of them. As shown in Figure 3a, it can be found that modifying the value of the force field cannot solve this problem. No matter how the values of attraction force and repulsion force are modified, it can only aggravate the instability of the robot's motion, inducing vibration. This paper chooses to modify the direction of the attraction potential field to solve this problem. The general idea is as below: taking the goal as the center, a quadrilateral is generated within a certain distance. When the value of the resultant force is 0, the four corners of the created quadrilateral will be regarded as interim goals for navigation, and choosing which one is the goal relies on the evaluation function. After the robot reaches a position at a certain distance \(d_{0}\) from the goal, the four virtual goals will be invalidated, so that the robot can reach the real goal. In addition, if the target point cannot be reached, it means the goal is too close to the obstacles, so we could increase the attraction force to make it reach the goal. According to the traditional attraction  
Figure 4: The goal unreachable problem.  
Figure 3: The local minimum problem. (**a**) Case 1. (**b**) Case 2.  
potential function, the attraction force is inversely proportion to the distance between the robot and the goal. Therefore, we can assume the attraction force as a constant within a certain range \(d_{0}\) from the goal. In order to avoid the situation where the value of the resultant force is still 0 when the robot is near the goal, we can increase \(d_{0}\) to solve this problem. The improved attraction potential field (Equation (17)) and schematic diagram (Figure 5) are as follows:  
\[U_{att}(q)=\left\{\begin{array}{l}\frac{1}{2}k_{att}\left(q-q_{g}\right)^{2},\ U\
eq 0\,\rho(q)\ \geq\ d_{0}\\ \frac{1}{2}k_{att}\left(q-q_{gi}\right)^{2},\ U=0\,\rho(q)\ \geq\ d_{0}\\ \frac{1}{2}k_{att}d_{0}^{2},\ U\ \
eq\ 0\,\rho(q)<d_{0}\end{array}\right. \tag{17}\]  
Equation (18) mainly compares the angle value \(\theta\) between attraction force and repulsion force. When the value of the resultant force is 0, the angle value is equal to 180\({}^{\circ}\). As such, the greater the difference is between the real angle value and 180\({}^{\circ}\), the higher the evaluation function value's virtual goal can be; the robot is much more likely to move in the direction of the virtual goal a with higher evaluation function value.  
\[C(q)=\mid \tag{18}\]