# Dynamic Heterogeneous Search-Mutation Structure-Based Equilibrium Optimizer
## 5 DHSMEO for UAV Mountain Path-Planning Problem
### Problem Description  
It is necessary to clarify the UAV mountain path-planning issue for achieving the mission goal in UAV mountain path planning. Missions like disaster rescue and military reconnaissance often involve the UAV mountain path-planning problem. The UAV mountain path-planning problem requires generating the minimum-distance path connecting the starting position and the ending position while avoiding the infeasible obstacle area and the threatening dangerous area. This ensures that the UAV can safely complete the assigned mission along the planned path with the lowest energy consumption. To achieve the above goal and validate the practicality of DHSMEO in UAV mountain path-planning issue, the three-dimensional mountain map associated with this problem needs to be constructed at first. According to the stated goal, the three-dimensional mountain map needs to account for the infeasible mountain terrain and the threatening dangerous area. The model used to generate the mountain terrain in the three-dimensional mountain map is given via the following equation: \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **EO** & **RSA** & **SMA** & **HGS** & **DMMAEO** \\ \hline [PERSON] mean rank & 6.487 & 8.526 & 6.077 & 6.410 & 3.462 \\ Final rank & 8 & 10 & 6 & 7 & 2 \\
1/0/\(-1\) & 32/6/1 & 30/7/2 & 31/5/3 & 30/6/3 & 27/10/2 \\ \hline  & **DEEO** & **HRSA** & **ESMA** & **AOAHGS** & **DHSMEO** \\ \hline [PERSON] mean rank & 4.821 & 7.192 & 4.231 & 5.538 & **2.256** \\ Final rank & 4 & 9 & 3 & 5 & **1** \\
1/0/\(-1\) & 28/9/2 & 32/7/0 & 31/5/3 & 31/5/3 & \(\sim\) \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Non-parametric test results for DHSMEO and compared algorithms in 39 benchmark functions. \[z=\sum_{u=1}^{u}h_{u}\exp[-(\frac{x-x_{u}}{x_{\rm p(}u)})^{2}-(\frac{y-y_{u}}{y_{ \rm p(}u)})^{2}], \tag{23}\]  
where the following applies: \(z\) represents the height in a three-dimensional landscape; \(x\) and \(y\) represent the coordinates for horizontal points in a three-dimensional landscape; \(U\) is the number of peaks in the three-dimensional terrain; \(x_{u}\) and \(y_{u}\) are the coordinates of the \(u\)-th mountain peak; \(h_{u}\) denotes the mountain contour of the \(u\)-th mountain peak; and \(x_{\rm p(}u)\) and \(y_{\rm p(}u)\) are the mountain contour parameters of the \(u\)-th mountain peak. The threatening, dangerous area in the three-dimensional mountain map is modeled as a cylindrical region, which is used to represent the dangerous area commonly encountered in UAV mountain path-planning missions, such as the high-temperature fire zone in disaster rescue and the radar detection zone in military reconnaissance. The model used to generate the threatening, dangerous area is shown in the following equation:  
\[Threat_{v}=(x_{v},y_{v},z_{v},R_{v}), \tag{24}\]  
where \(Threat_{v}\) represents the \(v\)-th dangerous cylindrical region; \(x_{v}\) and \(y_{v}\) denote the horizontal and vertical coordinates for the center within the \(v\)-th dangerous cylindrical area, respectively; \(z_{v}\) represents the height of the \(v\)-th dangerous cylindrical area; and \(R_{v}\) represents the radius of the \(v\)-th dangerous cylindrical area. Compared to the infeasible mountain terrain, the interior of the dangerous cylindrical area is designated as infeasible, while its neighborhood area remains feasible but threatening. Based on the aforementioned description of the UAV mountain path-planning problem, it is expressed with the below minimization problem to meet the mission requirements:  
\[ \begin{array}{ll}\min&J_{\rm UM}=\|\mathbf{Q}\|+\eta_{\rm T}J_{\rm T},\\ \rm s.t.&\mathbf{Q}\
otin O_{\rm b},\\ \\ \end{array}  \tag{25}\]  
\[J_{\rm T}=\sum_{k=1}^{K-1}\sum_{v=1}^{V}J_{\rm T(v,k)}, \tag{26}\]  
\[J_{\rm T(v,k)}= \begin{cases}0,&d_{(v,k)}>D_{v}+R_{v},\\ D_{v}+R_{v}-d_{(v,k)},&R_{v}<d_{(v,k)}\leq D_{v}+R_{v},\\ \infty,&d_{(v,k)}\leq R_{v}.\end{cases}