# UAV Path Planning Model Based on R5 DOS Model Improved A-Star Algorithm
## 2 Materials and Methods  
### UAV Representation Based on R5 DOS-Intersection Model  
In 2021, [PERSON] et al. [22] proposed a multi-UAV formation based on the R5 DOS intersection model. Improvements were made to the R5 DOS intersection model. Comparing the R5 DOS model formation with the rectangular model and the grid model, it can be seen that the R5 DOS model formation can effectively reduce communication, time and energy costs, while enhancing the robustness of the UAV formation and ensuring the integrity of the UAV's formation communication. Simulation experiments were carried out on the model and the results showed that the model formation was stable. Additionally, when the leader failed, the follower could quickly replace the leader and improve the robustness of the formation.  
The R5 DOS model divides the space into 16 regions and combines topological relationships:\[R5 DOS=\begin{pmatrix}\varepsilon(A^{\circ}\cap B^{\circ}\cap C^{\circ})& \varepsilon(A^{\circ}\cap B^{\circ}\cap(C^{\circ})^{\circ})&\varepsilon(A^{ \circ}\cap(B^{\circ})^{\circ}\cap C^{\circ})&\varepsilon(A^{\circ}\cap(B^{ \circ})^{\circ}\cap(C^{\circ})^{\circ})\\ \varepsilon((A^{\circ})^{\circ}\cap B^{\circ}\cap C^{\circ})&\varepsilon((A^ {\circ})^{\circ}\cap B^{\circ}\cap(C^{\circ})^{\circ})&\varepsilon((A^{ \circ})^{\circ}\cap(B^{\circ})^{\circ}\cap C^{\circ})&\varepsilon((A^{\circ} )^{\circ}\cap(B^{\circ})^{\circ}\cap(C^{\circ})^{\circ})\\ \varepsilon(s1 NE)&\varepsilon(s2 NE)&\varepsilon(s3 NW)&\varepsilon(s4 NW)\\ \varepsilon(s1 EN)&\varepsilon(s2 EN)&\varepsilon(s3 NW)&\varepsilon(s4 NW)\\ \varepsilon(s5 ES)&\varepsilon(s6 ES)&\varepsilon(s7 WS)&\varepsilon(s8 WS)\\ \varepsilon(s5 SE)&\varepsilon(s6 SE)&\varepsilon(s7 SW)&\varepsilon(s8 SW)\\ \end{pmatrix} \tag{1}\]  
Here,  
\[\begin{cases}s1 NE;x_{b}\geq 0,y_{b}\geq 0,z_{b}\geq 0,\theta_{ob}\in\left[0, \frac{\pi}{4}\right)\\ s2 NE;x_{b}<0,y_{b}\geq 0,z_{b}\geq 0,\theta_{ob}\in\left[0,\frac{\pi}{4}\right)\\ s1 EN;x_{b}\geq 0,y_{b}\geq 0,z_{b}\geq 0,\theta_{ob}\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\\ s2 EN;x_{b}<0,y_{b}\geq 0,z_{b}\geq 0,\theta_{ob}\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\\ s5 SE;x_{b}>0,y_{b}\geq 0,z_{b}<0,\theta_{ob}\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\\ s6 SE;x_{b}<0,y_{b}\geq 0,z_{b}<0,\theta_{ob}\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\\ \end{cases}\begin{cases}s5 ES;x_{b}\geq 0,y_{b}\geq 0,z_{b}<0,\theta_{ob}\in\left[ \frac{\pi}{2},\frac{\pi}{4}\right)\\ s6 ES;x_{b}<0,y_{b}\geq 0,2z_{b}<0,\theta_{ob}\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\\ s5 SE;x_{b}>0,y_{b}\geq 0,z_{b}<0,\theta_{ob}\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\\ s6 SE;x_{b}<0,y_{b}\geq 0,z_{b}<0,\theta_{ob}\in\left[\frac{3\pi}{2}, \frac{\pi}{4}\right)\\ \end{cases} \tag{2}\]  
\[\begin{cases}s8 SW;x_{b}\geq 0,y_{b}\geq 0,z_{b}<0,\theta_{ob}\in\left[\pi, \frac{5\pi}{4}\right)\\ s7 SW;x_{b}\geq 0,y_{b}\geq 0,z_{b}\geq 0,\theta_{ob}\in\left[\pi,\frac{5\pi}{4} \right)\\ s8 WN;x_{b}<0,y_{b}<0,z_{b}\geq 0,\theta_{ob}\in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\\ s8 WS;x_{b}\geq 0,y_{b}<0,z_{b}<0,\theta_{ob}\in\left[\frac{5\pi}{4}, \frac{\pi}{2}\right)\\ s7 WS;x_{b}<0,y_{b}<0,z_{b}<0,\theta_{ob}\in\left[\frac{5\pi}{4}, \frac{3\pi}{2}\right)\\ s7 WS;x_{b}<0,y_{b}<0,z_{b}<0,\theta_{ob}\in\left[\frac{5\pi}{4  
**Definition 2**.: _For the DOS layer, we define in Formula 3._  
\[\varepsilon(DOS)=\begin{cases}0:\text{ No objects in this area}\\ 1:\text{ Obstacles in this area}\end{cases} \tag{3}\]  
The improvements to the A-star algorithm in this paper are based on this model.