# Research on Airspace Conflict Detection Method Based on Spherical Discrete Grid Representation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Determination of the Airspace Grid Subdivision System

#### 2.1. Introduction of Typical Global Subdivision Schemes

- (1)
- Equal longitude and latitude grid

- (2)
- Variable longitude and latitude grid

- (3)
- Adaptive spherical grid

- (4)
- Regular polyhedral grid

#### 2.2. Comparison of Airspace Grid Subdivision Systems

## 3. Construction of Airspace Grid Model Based on Icosahedron Spherical Rhombus Subdivision

#### 3.1. Construction of Subdivision Model

#### 3.2. Interchange of Coordinate Systems

- (1)
- Conversion of longitude and latitude coordinates to spherical Cartesian coordinates;

- (2)
- Conversion of spherical Cartesian coordinates to longitude and latitude coordinates;

#### 3.3. Grid Coding Method

- The second part: location code: the 1–3rd levels of segmentation. The grid quadtree is divided, the grid space is filled by the Hilbert curve, and the binary code is converted into Hilbert code;
- The third part: coordinate code: the 4–10th levels of segmentation. The grid quadtree is divided, and the grid space is filled by the Hilbert curve. The binary code is converted into Hilbert code, and then the Hilbert code is converted into (i, j) coordinates.

## 4. Algorithm Model Construction

#### 4.1. Airspace Representation

- (1)
- Element information representation
- Working area: the area set by the airspace is set as a working area. The scope of the working area is composed of 2n adjacent diamond areas of the same level as required. This paper takes a diamond area of the third level as an example, with an area of about 800,000 km
^{2}. The number of the work area is represented by the 1–3rd levels of the grid, that is, the location code; - Airspace: It is represented by the grid matrix at levels 4–10th. The grid surrounded by the four points of the coordinates of the boundary points of the airspace is the occupied grid of the airspace. Each airspace occupies the grid and expands one layer of the grid to obtain the airspace envelope. The initial 10 rhombus are level 0. According to the calculation formula of spherical arc length, the length of a single rhombus edge is about 7061 km. When dividing to level 10, the length of a single rhombus edge is about 7 km. The minimum distance of non-adjacent rhombus grids is one rhombus grid interval, that is, the vertical distance from the apex of the rhombus to the opposite side is about 6 km. During CD, the airspace is extended by one layer, that is, two surrounding layers are separated, and the minimum distance is 11.94 km, greater than 10 km. Therefore, the division to the 10th level not only meets the minimum safe separation distance of airspace, but also does not cause grid deformation in different latitude areas, and meets the accuracy conditions required for the use of the AR grid;
- Route: the width of the air route is 20 km, and the minimum is not less than 8 km. Therefore, the route can be regarded as a linear airspace model. As shown in Figure 10.

- (2)
- Construction of airspace data matrix
- (a)
- The longitude and latitude coordinates of airspace boundary points are converted into spherical rectangular coordinates. The longitude and latitude coordinate G (longitude λ, latitude φ), The equatorial radius of the Earth R corresponds to the three-dimensional rectangular coordinates C(X, Y, Z). The relationship of transformation from longitude and latitude coordinates to spherical rectangular coordinates can be established from the spatial geometric relationship, as shown in Figure 5.
- (b)
- Spherical rectangular coordinate grid positioning. Given the grid vertex coordinates P
_{1}, P_{2}, P_{3}, P_{4}and the coordinates of the point to be solved, connect the midpoint on the opposite side of the diamond grid to obtain the midpoint coordinates M_{1}, M_{2}, M_{3}, M_{4}. As shown in Figure 8, divide the grid into four quadrants, calculate the plane OM_{1}M_{3}normal vector $\overrightarrow{{v}_{1}}$ and the plane OM_{2}M_{4}normal vector $\overrightarrow{{v}_{2}}$, and determine the quadrant of the point according to the relationship between the point and the plane. The initial state of Hilbert is State I. The Hilbert state of the next level is determined according to the quadrant transfer mode. Carry out hierarchical division until the 10th level. As shown in Figure 11. - (c)
- Airspace matrix construction. After connecting the four points of the coordinate code, the boundary of the airspace is obtained, the grid occupied by the airspace is calculated, and the value is assigned to 1, and the value of other grids in the mission area is assigned to 0, and a 128 × 128 data matrix composed of 0 and 1 is obtained.
- (d)
- Determination of airspace occupancy grid. Connect the coordinates of the airspace boundary points to obtain the airspace polygon, and determine whether all points in the matrix are within the airspace range. The value is assigned to 1 in the airspace range, and the value is 0 in the non-domain. The 128 × 128 matrix composed of 0.1 is the airspace data matrix.

#### 4.2. Conflict Detection Method on the Base of Matrix Analysis

_{ij}} and B = {b

_{ij}} are two matrices of the same order, if c

_{ij}= a

_{ij}× b

_{ij}, then the matrix C = {c

_{ij}} is the Hadamard product of A and B, recorded as $A\circ B$.

_{ij}, and other grids in the work area are assigned a value of 0, and the data matrix of airspace A, $matA$ = {a

_{ij}} is obtained; The grid occupied by airspace B is assigned a value of bij, and other grids in the work area are assigned a value of 0. The data matrix of airspace B, $matB$ = {b

_{ij}} is obtained 128 × 128.

_{ij}= a

_{ij}× b

_{ij}, it shows that the grid in row i and column j is occupied by airspace A and B at the same time, and the exact location of conflict is detected; if $matC={O}_{\left(i\times j\right)}$, there is no conflict.

#### 4.3. Conflict Detection Model on the Base of Airspace Analysis

- Step 1 time attribute CD. Compare the start time and end time of airspace A with other airspace. If there is a conflict with the intersection of the empty time, put the airspace An and the airspace with which there is a conflict into an airspace set;
- Step 2 height attribute CD. Logical relationship and time attribute detection: compare whether there is an intersection in the airspace altitude layer. After the CD of time and height dimensions is completed, several airspace sets are obtained. There are both time conflicts and high conflicts in the airspace within these airspace sets;
- Step 3 range attribute CD. Obtain the data matrix of airspace A and the data matrix of airspace B. Carry on the Hadamard product operation $matA\circ matB$ to obtain the $matC$ of the same dimension. At that time $matC={O}_{\left(i\times j\right)}$, it is determined that there is no conflict between airspace A and airspace B;$$\left[\begin{array}{ccccc}1& 1& \cdots & 0& 0\\ 1& 1& \cdots & 0& 0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1& 1& \cdots & 0& 0\\ 1& 1& \cdots & 0& 0\end{array}\right]\circ \left[\begin{array}{ccccc}0& 0& \cdots & 1& 1\\ 0& 0& \cdots & 1& 1\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0& 0& \cdots & 1& 1\\ 0& 0& \cdots & 1& 1\end{array}\right]=\left[\begin{array}{ccccc}0& 0& \cdots & 0& 0\\ 0& 0& \cdots & 0& 0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0& 0& \cdots & 0& 0\\ 0& 0& \cdots & 0& 0\end{array}\right]$$
- Step 4 outputs the airspace CD results. Starting from the high-level airspace, it is detected in accordance with steps 1–3, and cycle to determine whether all airspace CD is completed.

## 5. Simulation Experiment

- (1)
- Workspace and AR

- (2)
- Perform airspace 3D CD

- (3)
- Simulation conclusion and analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AG | Airspace Grid |

AR | Airspace Representation |

CD | Conflict Detection |

GIS | Geographic Information System |

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Subdivision Level | Rhombus Side Length/km | Number of Global Grids | Rhombic Spherical Area/km^{2} | Distance from the Vertex of the Rhombus to the Opposite Side/km |
---|---|---|---|---|

0 | 7061 | 10 | 51,006,786 | 6115 |

1 | 3630 | 40 | 12,751,696 | 3057 |

2 | 1865 | 160 | 3,187,924 | 1529 |

3 | 960 | 640 | 796,981 | 765 |

4 | 480 | 2560 | 199,245 | 382 |

5 | 221 | 10,240 | 49,811 | 191 |

6 | 110 | 40,960 | 12,452 | 95 |

7 | 55 | 163,840 | 3113 | 47 |

8 | 28 | 655,360 | 778 | 24 |

9 | 14 | 2,621,440 | 194 | 12 |

10 | 7 | 10,485,760 | 48 | 6 |

Vertice | Longitude | Latitude | Vertice | Longitude | Latitude |
---|---|---|---|---|---|

S0 | 0° E | 90° N | S6 | 36° E | 26.57° S |

S1 | 0° E | 26.57° N | S7 | 108° E | 26.57° S |

S2 | 72° E | 26.57° N | S8 | 180° E | 26.57° S |

S3 | 144° E | 26.57° N | S9 | 108° W | 26.57° S |

S4 | 144° W | 26.57° N | S10 | 36° W | 26.57° S |

S5 | 72° W | 26.57° N | S11 | 0° E | 90° S |

Longitude and Latitude Coordinate | Global Code | Diamond Code | Location Code | Coordinate Code |
---|---|---|---|---|

[113.46, 30.30] | [1-32-(52, 74)] | 1 | 32 | (52, 74) |

[112.95, 29.77] | [1-32-(42, 73)] | 1 | 32 | (42, 73) |

[113.57, 29.43] | [1-32-(47, 83)] | 1 | 32 | (47, 83) |

[113.82, 29.99] | [1-32-(54, 81)] | 1 | 32 | (54, 81) |

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**MDPI and ACS Style**

Qu, K.; Zhao, G.; Wu, Y.; Tong, L.
Research on Airspace Conflict Detection Method Based on Spherical Discrete Grid Representation. *Appl. Sci.* **2023**, *13*, 6493.
https://doi.org/10.3390/app13116493

**AMA Style**

Qu K, Zhao G, Wu Y, Tong L.
Research on Airspace Conflict Detection Method Based on Spherical Discrete Grid Representation. *Applied Sciences*. 2023; 13(11):6493.
https://doi.org/10.3390/app13116493

**Chicago/Turabian Style**

Qu, Kai, Guhao Zhao, Yarong Wu, and Liang Tong.
2023. "Research on Airspace Conflict Detection Method Based on Spherical Discrete Grid Representation" *Applied Sciences* 13, no. 11: 6493.
https://doi.org/10.3390/app13116493