# Global Time-Varying Path Planning Method Based on Tunable Bezier Curves
## 3 Transient Path under the Single-Obstacle Environment
### Obtaining Initial Information  
In Step 1, the kinematic model of the snake manipulator is constructed, including establishment of the D-H coordinate system and analysis of the conversion relationship between the parameters. According to the task requirements, the start point \(x_{\text{start}}\), the end point \(x_{\text{end}}\), the start point direction \(d_{\text{start}}\) and the end point direction \(d_{\text{end}}\) are set corresponding to the motion trajectory of the manipulator. The relationship between the  
Figure 4: Bezier curve generated for irregularly shaped obstacles.  
Figure 3: Bezier curve generated for non-parallelogram obstacles.  
Figure 2: Bezier curve generated for circular obstacles.  
parameters is presented in Equation (1). The \(d_{\text{start}}\) and \(d_{\text{end}}\) in Figures 1, 2 and 4 are both \(0^{\circ}\), so they are not marked. The \(d_{\text{start}}\) and \(d_{\text{end}}\) in Figure 3 are \(-30^{\circ}\).  
\[\begin{cases}d_{\text{start}}=x^{\prime}\,_{\text{start}}\\ d_{\text{end}}=x^{\prime}\,_{\text{end}} \tag{1}\]  
In Step 2, the detailed information of obstacles can be obtained through image processing, construction of sparse maps and so on. The feature information of obstacles (represented by the Obs_Information symbol) can be extracted and only one set of data is needed to store the shape data of obstacles: Obs_Information (1). The data of \(x\), \(y\), \(z\), \(\theta_{x}\), \(\theta_{y}\) and \(\theta_{z}\) are used for storage of the position data of three-dimensional obstacles: Obs_Information (2:7). Where \(x\), \(y\) and \(z\) are the distance between the current position and the origin point, \(\theta_{x}\), \(\theta_{y}\) and \(\theta_{z}\) are the angles of rotation of the initial pose around the axes \(x\), \(y\) and \(z\). The position data of two-dimensional obstacles are stored through the data of \(x\), \(y\) and \(z\). The data of \(z\), \(\theta_{x}\) and \(\theta_{y}\) are always zero. Size data of the obstacle are stored through obstacle features: Obs_Information (8:), the size data of obstacles of different shapes occupy different numbers of data. For example, the size data of spherical obstacles are stored through \(r\), the size data of cuboid obstacles are stored through \(a\), \(b\) and \(c\). The size data of irregular polyhedral obstacles are stored through the initial position information of each vertex.  
The problems of image recognition and segmentation can be solved by existing methods proposed by other researchers. For obstacle recognition, [PERSON] [28] proposed an adaptive object recognition system, which can effectively identify specific targets under complex backgrounds. For the extraction of edge information, [PERSON] [29] used the improved wavelet mode maximum algorithm to extract image edges, which can obtain edge image information with better clarity and connectivity. [PERSON] [30] extracted the boundary of an obstacle from the semantic segmentation result by applying pixel filtering. For irregular obstacles, [PERSON] [31] conducted grid preprocessing and convex preprocessing for concave obstacles, which enhanced the safety of UAV path obstacle avoidance. In order to determine turning points, [PERSON] [32] proposed to use motion coherence to distinguish dynamic and static visual feature points and remove the edges between irrelevant points in the point correlation optimization process.  
The shape and size of obstacles do not change with time, but the position of dynamic obstacles changes over time, such as translation, rotation and other movements. Therefore, it is necessary to extract the information of dynamic obstacles in real time to obtain preliminary data for obstacle avoidance.