## 2 Global Destination Arrival Sequence Planning
### Multi-Destination Ranking Method Based on Environmental Obstacle Value  
Definition: Robustness means that the results should not be greatly affected by data disturbances, noise and outliers present in the algorithm. The simulated annealing algorithm [19] has few parameters, short calculation time, and robustness. It is suitable for the high efficiency requirements of this article. The multi-destination sorting process adopts the simulated annealing algorithm to optimize the sorting. Steps of the simulated annealing multi-destination sorting algorithm based on the environmental obstacle value are as follows. Step 1. The initial temperature \(Te_{max}\), the end temperature \(Te_{min}\) and the cooling rate \(r=0.8\) (\(0<r<1\)) are determined as Equation (3). The starting point and \(n\) destination points form the matrix (\(S,\)\(T_{1},\)\(T_{2},\)..., \(T_{n},\)\(S\)). \[Te_{max}=\sqrt{\sum\limits_{i=1}^{n}G_{i}^{2}}\qquad Te_{max}-Te_{min}=n\times Te _{max}(1-r)\qquad, \tag{3}\]  
where \(G_{i}\) is the \(i\)-th environmental obstacle value. Step 2. The elements in the matrix (\(S,\)\(T_{1},\)\(T_{2},\)..., \(T_{n},\)\(S\)) are rearranged to form the solution spaces \(I^{n!\times(n+2)}\) as shown in the Equation (4). The number of solutions in the solution spaces \(I^{n!\times(n+2)}\) is \(n!\). \[I=\left( \begin{array}{ccccc}S&T_{1}&T_{2}&\cdots&T_{n}&S\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ S&T_{2}&T_{1}&\cdots&T_{n}&S\\ S&T_{n}&T_{1}&\cdots&T_{n-1}&S\\ \end{array} \right), \tag{4}\]  
Additionally, the solution space \(I\) randomly generates an initial solution \(I_{0}\) (\(I_{0}\)\(\in\)\(I\)). Step 3. Generate a new solution through the two-transformation method. The selected sequence number is m, k. The access sequence between the \(m\)-th and \(k\)-th solution elements is exchanged. If the previous path solution of the exchange is \(I_{0}\) = (\(S,\)\(T_{1},\)\(T_{2},\)..., \(T_{m},\)..., \(T_{k},\)..., \(T_{n},\)\(S\)), the exchanged path is a new path, as shown in Equation (5). \[I^{\prime}_{i}=\big{(} \begin{array}{ccccc}&S&T_{1}&T_{2}&\cdots&T_{k}&\cdots &T_{m}&\cdots&T_{n}&S\\ \end{array} \big{)}, \tag{5}\]  
Step 4. This step sets the environmental obstacle value as the objective function. The objective function is \(f(S,\)\(T_{1},\)\(T_{2},\)\(\ldots,\)\(T_{n},\)\(S)\). The difference between the objective function of solution \(I_{0}\) before transformation and that of solution \(I^{\prime}_{i}\) after transformation is calculated from Equations (6) and (7). \[f(S,T_{1},T_{2},\ldots,T_{n},S)=\sum\limits_{i=1}^{n+2}d(c_{i},c_{i+1}), \tag{6}\]\[\Delta f=f(I_{i}^{\prime})-f(I_{i}), \tag{7}\]  
where the difference between the objective function of the solution before transformation and that of solution after transformation is \(\Delta f\). \(c_{i}\) and \(c_{i+1}\) are two adjacent target points, respectively. \(d(c_{i},c_{i+1})\) is the obstacle value of two adjacent target points. Definition: accepting a new state with probability instead of using completely determined rules is called Metropolis criterion. Step 5. According to Equation (8), the acceptance probability \(p\) is calculated. The Equation (8) accepts the rule of Metropolis criterion to select the next generation of new solutions. If \(\Delta f<0\), it means that the cost of the solution after transformation is less than the solution before transformation, so the new solution is \(\mathbf{I}_{i}\). If \(\Delta f>0\), it means that the cost of the solution after transformation is greater than the solution before transformation. The new solution is judged by accepting the current path as the new path probability \(p\). The greater the probability \(p\), the greater the possibility of acceptance. \[p= \begin{cases}1&(\Delta f<0)\\ \exp(-\Delta f/Te)&(\Delta f>0)\end{cases} , \tag{8}\]  
This step has a certain probability to accept the solution that is worse than the current solution, so it can jump out of the local optimum to a certain extent.