## 3 Method
### Procedures of AOQPIO  
Differing from other conventional PIO algorithms, the process of the proposed method in this paper has mainly three improvements. Firstly, the chaotic strategy is introduced in our algorithm to generate some initial solutions with wider solution space coverage. Further, the map and compass factor \(R\), varying with the number of iterations, is adopted to balance global search ability and local search ability. Last but not least, the new population updating equation is proposed to increase the search diversity at the landmark operator period. Overall, the complete procedure of AOQPIO algorithm can be presented as the following Algorithm 1:  
```
//*Initialization*/
1 Set initial values for \(D\), \(N_{p}\), \(N_{c1\max}\), \(N_{c2\max}\), \(q\), \(w\), \(N_{dec}\);
2 Generate the chaos sequence \(U\) according to Equation (20);
3 Set initial path \(X_{i}=[ \begin{array}{cccc}X_{i,1}&X_{i,2}&\cdots&X_{i,D}\end{array} ]\), velocity \(V_{i}=[ \begin{array}{cccc}V_{i,1}&V_{i,2}&\cdots&V_{i,D}\end{array} ]\) for pigeon according to Equation (21);
4 Calculate fitness values of different pigeon individuals, Set \(X_{pi}=X_{i}\), \(X_{g}=\operatorname{argmin}[calculate\_fitness(X_{pi})]\); /*Map and compass operations with quantum-behavior*/
5 for\(N_{c}=1\):\(N_{c1\max}\)
6 Generate parameter \(\varphi\), \(\varepsilon\), \(m\);  Update parameter \(R\) according to Equations (22) and (23);
8 Sum the \(X_{pi}(i)\) for each \(i\);
10 Calculate m_best according to Equation (15);
11 for\(i=1\):\(N_{p}\)
12 Calculate \(P_{i}\) according to Equation (14);
13 Update \(V_{i}\) according to Equation (11);
14 Update \(X_{i}\) with quantum-behavior according to Equation (10);
15 new_fitness = calculate_fitness(\(X_{i}(N_{c})\));
16 If new_fitness < Fitness(\(i\)) then Fitness(\(i\)) = new_fitness; \(X_{pi}\)=\(X_{i}\);
18 endif
19\(X_{g}=\operatorname{argmin}[calculate\_fitness(X_{pi}(N_{c}))]\);
20 endfor
21 endfor
22 /*Landmark operator*/
24 for\(N_{c}=N_{c1\max}\)+1:\(N_{c2\max}\)
25 Rank all the available pigeon individuals according to their fitness values;
26 Update the population of pigeon \(N_{p}\) according to Equation (24);
27 Calculate the center of the pigeon \(X_{c}(N_{c})\) according to Equation (18);
28 for\(i=1\):\(N_{p}\)
29 Update \(X_{i}\) according to Equation (26);
30 Evaluate \(X_{i}\), and update \(X_{pi}\) and \(X_{g}\) according to line (16)-(20);
31 endfor
32 endfor
33 /*Output*/
34\(X_{g}\) is output as the global optimal of the fitness function

### AOQPIO-Based UAV Path Planning

In order to further illustrate AOQPIO, in this paper, to solve UAV path planning problem, the detailed steps of AOQPIO-based UAV path planning are described in this section. If the user-defined number of control points of the UAV path is \(D\), then the path planner should determine \(D1=3*D\) coordinate parameters, namely \(X_{1},X_{2},\cdots,X_{D1}\), where \((X_{i},X_{D+i},X_{2D+i})\), \(i=1,2,\cdots,D\) denote the three dimensional coordinates in solution space. The point \((x_{i},x_{D+i},x_{2D+i})\) in solution space can be transformed into the point \((x(i),y(i),z(i))\) in the real flying space by the following transform equation [41]:

\[\left[ \begin{array}{c}x(i)\\ y(i)\\ z(i)\end{array} \right]=\left[ \begin{array}{ccc}\cos\xi&\sin\xi&0\\ -\sin\xi&\cos\xi&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}x_{i}\\ x_{D+i}\\ x_{2D+i}\end{array} \right]+\left[ \begin{array}{c}x_{start}\\ y_{start}\\ z_{start}\end{array} \right], \tag{23}\]

where the \((x_{start},y_{start},z_{start})\) represents the coordinate of the given start point in the flying space, and the \(\xi\) is the angle that the x-axis in solution space counterclockwise rotate to parallel segment start and goal point.