## 2 Methods
### Path-Planning Improvement
#### 2.2.1 Fan-Shaped Goal Orientation  
As a sampling-based path-planning method, the selection of sampling points has a great influence on the efficiency of the RRT algorithm. The selection of sampling points in the traditional RRT algorithm is completely random, so it has the disadvantages of blind search, low search efficiency, and being unable to handle complex environments. This study enhances the efficiency of searches by refining the selection process of sampling points through the application of a fan-shaped goal orientation approach to enlarge the random tree. The fan-shaped goal orientation strategy takes the goal point as the reference point of the random tree expansion direction, so that each sampling can be closer to the goal point, effectively preventing the reverse search of random tree nodes, ensuring the correct direction of sampling and improving the search efficiency. Concurrently, sampling points are chosen at random within the fan-shaped area, maintaining the randomness in point selection, enabling it to seek a viable route when facing obstacles. This unique strategy significantly reduces the number of invalid sampling points caused by random sampling. And when the robot is in a complex environment, it can also make the robot quickly leave the dangerous area. This strategy not only greatly proves the efficiency of the algorithm but also ensures the adaptability of the algorithm to complex environments. The principle of the fan-shaped target orientation strategy is shown in Figure 2.  
The sampling points are selected in the sector, and the selection of angle \(\theta\) is fitted by the truncated standard normal distribution. Truncated normal distribution is a normal distribution that limits the range of variables. As shown in (1), the nearer the goal point, the higher the possibility of selection. \[f(\theta)=\left\{ \begin{array}{cc}c\frac{1}{\sqrt{2\pi}}e^{-\frac{\theta^{ 2}}{2}},&-90^{\circ}\leq\theta\leq 90^{\circ}\\ 0,&\text{else}\end{array} \right. \tag{1}\]  
where \(c\) is the truncation parameter; the selection range \(\theta\) is \(-90^{\circ}\) to \(90^{\circ}\). Directly taking the goal point as the sampling point of the random tree expansion direction makes it easy to fall into a local optimum, so this paper selects the sampling point according to the probability distribution. Firstly, a semi-circular arc is made with the connection between the position and the goal point as the center of the arc, and then the sampling points are selected in the sector area according to the probability of cutting  
Figure 2: Sampling point selection principle diagram, where \(x_{mnd}\) is the sampling point, \(goal\) is the goal point, \(\theta\) is the offset angle, \(\alpha\) is the angle to the x-axis, and \(x_{new}\) is a new node obtained by step expansion with a certain step size. off the standard normal distribution. Then, the set step is extended in the direction of the sampling point to obtain a new extended node, \(x_{new}\). When \(x_{rand}\) is in the left half of the sector, the coordinates of the new node, \(x_{new}\), in the X and Y directions are as follows:  
\[ \begin{array}{l}X_{new}=X_{near}+step\times\cos(\theta+\alpha)\\ Y_{new}=Y_{near}+step\times\sin(\theta+\alpha)\end{array}  \tag{2}\]  
When \(x_{rand}\) is in the right half of the sector, the coordinates of the new node, \(x_{new}\), in the X and Y directions are as follows:  
\[ \begin{array}{l}X_{new}=X_{near}+step\times\cos(\alpha-\theta)\\ Y_{new}=Y_{near}+step\times\sin(\alpha-\theta)\end{array}  \tag{3}\]  
The goal-oriented approach aims to direct the extension of the random tree based on the target point, enhance the path-planning algorithm's effectiveness in intricate environments, and elevate the quality of the planning path. The distribution of computing resources is fine-tuned to enhance global optimization potential, cut down search durations, and offer robust assistance for robotic navigation in intricate environments.