Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 5097324, 10 pages

https://doi.org/10.1155/2018/5097324

## Judgement Theorems and an Approach for Solving the Constellation-to-Ground Coverage Problem

Correspondence should be addressed to Maocai Wang

Received 15 September 2017; Accepted 14 January 2018; Published 22 February 2018

Academic Editor: Raffaele Solimene

Copyright © 2018 Zhiming Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The satellite constellation-to-ground coverage problem is a basic and important problem in satellite applications. A group of judgement theorems is given, and a novel approach based on these judgement theorems for judging whether a constellation can offer complete single or multiple coverage of a ground region is proposed. From the point view of mathematics, the constellation-to-ground coverage problem can be regarded as a problem entailing the intersection of spherical regions. Four judgement theorems that can translate the coverage problem into a judgement about the state of a group of ground points are proposed, thus allowing the problem to be efficiently solved. Single- and multiple-coverage problems are simulated, and the results show that this approach is correct and effective.

#### 1. Introduction

Currently, satellite constellations technology are widely used in many practical applications, including communication, navigation, remote sensing, and science missions [1, 2], and these applications provide very important support to many fields, such as the observation of the land, sea and air [3, 4], disaster monitoring [5, 6], and resource exploration and development [7]. To meet the requirements of these applications, some typical problems encountered when designing satellite constellations have been studied by scholars. These problems include constellation design [8, 9], constellation performance evaluation [10], satellite-ground communication [11], and satellite scheduling [12, 13]. In these typical problems, the service of the constellation to the ground region, which depends on the visibility between the constellation satellite and the ground region, is a basic and important problem. This problem is called the constellation-to-ground coverage problem.

Solutions to the constellation-to-ground coverage problem aim to calculate the coverage region and other information about the ground regions that are covered by satellite constellations during a time period or at a point in time. This problem is essentially a problem involving Boolean operations on 2D graphs [14, 15], which is commonly encountered in the fields of computer graphics and geology. However, compared with the traditional problems involving the Boolean operations on 2D graphs, the constellation-to-ground coverage problem has some differences because of the following three aspects: (1) the background manifold is a spherical or ellipsoidal surface in three dimensions rather than a 2D Euclidean space, (2) the shape of ground regions and the view field of the sensor can be arbitrary, and (3) the temporal characteristics during long simulation periods frequently arise in calculating the coverage region of sensor. These three aspects make this problem more difficult than traditional Boolean operation problems.

The solution of the coverage problem depends on the configuration of the constellation and the coverage problem itself. Different constellation configurations, including the star constellation [16], the Walker-Delta constellation [17, 18], the rosette constellation [19], and the flower constellation [20], may have some particular configuration characteristics, such as spatial symmetry [21, 22], common-track [20], regressive orbits characteristic [23], and polar orbits [24]. Proper usage of these characteristics can greatly increase computational efficiency. From the aspect of the coverage problem, owing to the different requirements of the time and space factors for different practical applications, constellation-to-ground coverage problems are different. For navigation and communication applications, the timeliness requirement is critical; however, for some remote sensing applications such as resource exploration, the timeliness requirement is minimal. Thus, the concepts of continuous coverage problems [8, 25] and discontinuous coverage problems [24, 26] have been proposed by many scholars, and thus, different constellation-to-ground coverage problems have different characteristics and different strategies for achieving a solution.

Until now, to solve constellation-to-ground coverage problems, there have been three typical methods, including the grid point method [27], the triangle mesh subdivision method [17], and the street-of-coverage method [28]. The triangle mesh subdivision method is a method that is favored by many scholars [18, 19]. This method can quickly judge whether the constellation can cover the globe at a given time. However, more detailed coverage information, such as the coverage rate, cannot be obtained. The street-of-coverage method is a method for solving the continuous coverage problem and can obtain the approximate analytic relationship between coverage and the constellation configuration so as to guide constellation design. The disadvantage of this approach is that the result is an approximate value and the method ignores phase information.

Additionally, some mathematical strategies have been applied in constellation-to-ground coverage problems, such as probability statistics [29], iterative methods [30], the differential quadrature method [31], the method based on the transformation group [21, 22], the polyhedron surrounding method [32], the method based on auxiliary space [26, 33], and the numerical fitting method [34]. These mathematical strategies can be used in some of the three aforementioned methods, which can accelerate computing efficiency. However, most of these mathematical strategies have some restrictions in that they can only work in some certain conditions and can only solve some certain problems.

In this paper, we focus on a subproblem of the constellation-to-ground coverage problem, the complete coverage problem, which judges whether a constellation can cover a ground region completely using one or multiple satellites. If the constellation cannot cover a ground region completely, we do not care about the coverage rate or other coverage properties.

In this paper, we will discuss this problem from the aspect of mathematics. From the point view of mathematics, the instantaneous coverage region of the ground region from a satellite can be seen as a spherical region. Thus, the constellation-to-ground coverage problem can be translated into a problem on the intersections of spherical regions. According to the basic characteristics of the coverage problem, a group of definitions are given, and a group of theorems as well as proofs are put forward. Then, according to the judgement theorems, highly accurate, efficient, and novel approaches for solving this problem are proposed.

The paper is organized as follows. Section 2 gives a brief review of the traditional methods. Section 3 introduces the basic concepts and appointments of this paper. Section 4 proposes a group of judgement theorems for this coverage problem. Section 5 discusses the detailed procedures of the approaches for solving this coverage problem. In Section 6, a group of experiments, including single-coverage problems and multiple-coverage problems, are conducted. Section 7 concludes the paper.

#### 2. Background

The traditional methods for solving the complete coverage problem include the grid point method and the triangle mesh subdivision method. A brief introduction to these two methods will be given.

##### 2.1. The Process of the Grid Point Method

The grid point method is one of the most commonly used approaches for solving the constellation-to-ground coverage problem. The main idea of this approach is discretization, which is the division of a continuous object into a group of discrete objects and then calculating the characteristics of these discrete objects to express the characteristics of the coverage problem.

For a coverage problem, two types of objects need to be discretized: the first is space, and the other is time. The discretization of space means discretizing the ground region into a group of grid points according to a regional discretization strategy, and the discretization of time means discretizing a time range into a set of time points.

For the grid point method, the process is as shown in Figure 1, and the basic steps are as follows: