# Global Time-Varying Path Planning Method Based on Tunable Bezier Curves
## 3 Transient Path under the Single-Obstacle Environment
### Feature Points of Obstacles  
In Step 4, the feature points of the obstacle are extracted. The turning point of each object is regarded as the feature point on the surface of each object. Several feature points and locations of feature points need to be extracted, which can be set according to the task, mainly related to the following factors.  
1. Obs_Information. The type of obstacle and the number of inflection points in the characteristic information of the obstacle;
2. The position of the start point \(x_{\text{start}}\) and the end point \(x_{\text{end}}\);
3. Which_Side. Whether the planned trajectory is above or below the obstacle needs to be determined; in other words, which side of the obstacle the planned trajectory bypasses needs to be determined.  
The pseudo-code corresponding to the Extract function is shown below.  
Line 2-3 in Algorithm 3. The \(E_{i}\) in Figure 1 represents the feature points of the rectangular obstacle, which are the points that characterize the shape and size of the obstacle. The fuzzy center of gravity \(C\) is determined according to the above feature points of the obstacle (also known as the inflection point of the obstacle), which can reduce the amount of calculation. Some specially shaped obstacles have no inflection point and several feature points of the obstacle can be set according to the specified rules, as shown in Figure 2. Where four straight lines with adjacent angles of \(60^{\circ}\) are made through the point \(O_{0}\) and the intersection point \(E_{i}\) of the straight line, and the intersection point with the circular obstacle is regarded as the feature point of the obstacle. By setting feature points for circular obstacles with the help of the method, the real-time generated Bezier curve never intersects with obstacles. The relevant proof process is shown in Section 5. Three or six feature points \(E_{i}\) can be extracted from triangular obstacles, four feature points \(E_{i}\) can be extracted from quadrangular obstacles, the number of feature points \(E_{i}\) extracted from \(N\)-sided obstacles is \(N\) and the number of feature points \(E_{i}\) extracted from circular obstacles is \(n\), where \(n\) can be set as an integer, such as 3-6, etc. After obtaining these feature points of the obstacle, the blurred center of gravity \(C\) of the obstacle can be obtained by taking the average of the above points. The calculation equation is as follows.  
\[C=\sum_{i=1}^{w}E_{i}/w \tag{5}\]  
where \(w\) is the number of feature points of the obstacle.  
```
1:Input: Obs_Information, \(x_{\text{start}}\), \(x_{\text{end}}\);
2:\(E_{i}\leftarrow\) Obs_Feature(Obs_Information);
3:\(C\leftarrow\) Equation (5)(\(E_{i}\));
4:Which_Side \(\leftarrow\) Judge_Side(\(C\), \(x_{\text{start}}\), \(x_{\text{end}}\));
5:Feature_Points \(\leftarrow\) (\(E_{i}\), Which_Side);
6:Output: Feature_Points;
```  
**Algorithm 3** Feature_Points \(\leftarrow\) Extract(Obs_Information, \(x_{\text{start}}\), \(x_{\text{end}}\))  
Line 4 in Algorithm 3. By determining which side of the connecting line between the start point \(x_{\text{start}}\) and the end point \(x_{\text{end}}\), the fuzzy center of gravity \(C\) is the side of the obstacle the moving object goes through can be determined. The fuzzy center of gravity \(C\), in Figures 1-4 is above the line, so the trajectory only needs to be planned in the upper part of the obstacle.