Formal System Construction And Solving Algorithm Research For Coverage Problem Of Constellation To Ground

With the rapid development and universalization of science and technology, space technology plays a more and more important role in military, political, economic and cultural fields, and becomes a critical part of the people’s daily life gradually. Its application relates to many fields, such as weather forecaste, navigational positioning, satellite timing service, military reconnaissance, communications, resources exploration, astronomical observations, and so on. Space technology can promote further development in many subjects, and lead the basic science and applied research of related discipline. Meanwhile, in modern warfare which is highly dependent on the support of air and space integrated system, the development of space technology has an important strategic deterrent on the military significance. The role of space industry in the future development of all countries will not only restricted to stimulate the development of domestic advanced science and technology and strengthen national defense capabilities, and also, it will decide the future pattern of the world to some extent.Coverage problem of constellation to ground is an essential problem in the applications of space technology. Coverage problem of constellation to ground involves a series of problems, and these problems can be classified into different categories according to different standards. According to satellite’s number of constellation, it can be classified into single satellite coverage problem and multiple satellites coverage problem. According to the type of ground target, it can be classified into point target coverage problem, line target coverage problem, area target coverage problem, latitude belt coverage problem and global coverage problem. According to the type of time set, it can be divided into the instantaneous coverage problem, continuous coverage problem, accumulative coverage problem and discontinuous coverage problems. According to the requirement of coverage level, it can be divided into single level coverage problem and multiple level coverage problem. According to the requirement of the result, it can be divided into judgment problem and calculating problem. For a concrete coverage problem, it is a permutation and combination of the above classification. The category of coverage problem of constellation to ground is various. For each certern coverage problem, experts and scholars have made the research, and put forward some algorithms for solving it. But these algorithms usually can only solve some certern coverage problems, and for other problems, it doesn’t work well. A uniform theoretical system which can analyze and describe all coverage problems is scarce, and a algorithm frame which can solve all coverage problems is also scare.The main work in this paper is establishing a complete and unified theory system for coverage problem of constellation to ground, making it can analyze and describe all types of the coverage problems in a unified form, and provide an algorithm framework that can solve all coverage problems and get a unified solution for coverage problem. In this paper, the research mainly includes four aspects of research work:1. The establishment of the theoretical system of coverage problem for constellation to ground and formal systemThe first major work of this paper is the establishment of the theoretical system of coverage problem for constellation to ground and formal system, that is the establishment of coverage state function, coverage area function and coverage time function.The first aspect of the first major work is the establishment of coverage state function. To make a comprehensive analysis of coverage problem of constellation to ground, we can get the following conclusion:coverage problem of constellation to ground involves three objects, that is satellite and ground targets and time. These three object called three elements of the coverage problem. Any coverage problem is a combination of three elements, and the three elements and its organization mode can only determine a coverage problem. From mathematical point of view, a covering problem can be regard as a map which independent variable is three elements. Then coverage problem can be seen as a function of the three elements. To analyze all of the coverage problems,we found that all coverage problems can be summed up in a primitive problem, that is whether the satellite can cover a ground point target at a certain time. This problem is named as the coverage state function. Because this coverage state function is only for single, single point and single moment, and in order to make this coverage function be appropriate for coverage problem as much as possible, the concept of the coverage state function is spread, and the independent variable parameters extend. Satellite parameter can be expanded from single satellite to constellation. Around target parameters can be expanded from point target to line target or area target. Time parameters can be extended from a single moment to an time range. When expanding, each parameter can expand in two kinds of way, existence extension and any extension. The different extension manner can corresponding to different organization form of three elements. After extending the coverage of state function, there still exist some cover problems that coverage state function cannot contain. So further expansion of the coverage function is need. According to multiple coverage problem, the satellite parameter have k-existence extension in addition to two usual extension mode. For intermittent coverage problem, it found that the time parameter has a expansion mode with the mixture of existence and any expansion mode. The existence mode mix any mode is called At time existence expansion mode, and the any expansion mode mix existence expansion mode is called At time any expansion mode. The latter mode corresponds to the intermittent coverage, while the former, also corresponds to the practical coverage problems, which is coverage problem with time length requirement.The second aspect of The first major work is the establishment of coverage area function and coverage time function. The space spanned by the three elements of cover state function can be seen as a manifold. The structure of the manifold is difficult to directly resolve, so projection method of ground target space and time parameters space is used. Projection to ground targets space can be seen as a mapping from Cartesian product of the three elements to ground target space, then this mapping is called coverage area function. Projection to time element space can be seen as a mapping from Cartesian product of the three elements to time element space, then this mapping is called coverage time function. On contract to coverage problem of constellation to ground target, coverage area function corresponds to the coverage range calculation problem or coverage rate calculation problem, while covering the time function corresponds to the time-varying problem and the time window calculation problem. According to this conclusion, it can be seen that the coverage problem and time window calculation problem have inner relation. They are different projections of the same manifold. These three functions above can represent almost all the coverage problem. Then all coverage problem can be formalized describe in a unified mode.When define various extensions of coverage state function, a set of symbol system is also defined. Each covering problems can use a special symbol to express. And also, each meaningful symbols correspond to a particular coverage problem. This set of symbols can be independently deduced, and can get some conclusion which direct analysis can not meet.2. The algorithm of solving any coverage problem.The second major work of this paper is to put forward a algorithm which can solve any coverage problem. To calculate problem of constellation to ground target, high price is need to pay when ask enough accurate solution. When solving coverage function, due to the complexity of cover function calculation, direct solution is not require, but rather, a function which result is always less than the coverage problem, which is called coverage infimum function, and a function which result is always greater than the coverage problem, which is called coverage supremum function, is need to seek. And also, in the ultimate sense, the three functions converge together. By solving the two cover function, a upper and lower bound of the coverage function can obtain. If the precision is less than the calculation problem requires, more accurate results can be obtained through more detailed division of the corresponding parameters.3. The analysis and solution of any coverage area function.The third major work of this paper is to analyze and solve coverage area function. The method for solving it is making an analysis of three main factors respectively, which is satellite, ground targets and time, and then make an comprehensive analysis, then an algorithm for solving any coverage function can obtain.The first step to analyze and compute coverage area function is to analyze the parameters of the satellite. The basic problem of covering problem is computing coverage range of satellite at a certain time. This is also the core part for solving the three cover functions. But this problem is a complicated problem, the complexity of the problem comes from three aspects:first, the shape of the sensor the satellite taken along is arbitrary. Second, the attitude of the satellite and the sensor is arbitrary; Third, the spherical projection function is difficult to analysis. Three aspects of mutual coupling, making the problem is very complex. In this paper, two special planes, which are called orthographic intersection plane and inclined intersection plane, turn this complicated problem into three simple problems. By this analysis, the invariant of the same satellite in different time and even different satellites can be found out. Thus reducing computational complexity. By imitating the method for using inscribed polygon and circumscribed polygons match a unresolved plane graph in plane geometry, a method for using inscribed spherical segment and circumscribed spherical segment match boundary of a unresolved sphere graph and using the spherical disc’s set operation to match coverage range of satellite to ground. Because spherical circle and spherical arc are both analytic object, we can use two simple analytical graphics from inside and outside to approach a graph which is hard to analytic. This two simple analytical graphics are called the infimum and supremum function of the coverage area function.The second step to analyze and compute coverage area function is to analyze the parameters of time. That is the calculation of continuous coverage, cumulative coverage and discontinuous coverage. These three issues are complex problems. The problem complexity of the three coverage problems is time parameter of the three problem is period, but in general, only the coverage range for a certain time can calculate. So, the method for solving the three problems is finding a set of infimum and supremum coverage functions which is denoted by a group of finite number of time. For continuous coverage problem, a suitable infimum and supremum coverage functions can be found, while for cumulative coverage problem, its lower bound function of natural existence, but the upper bound function is difficult to find. Therefore, the envelope’s upper bound of the range is used as the upper bound of the cumulative coverage range of functions. Intermittent problem is a compound of continuous coverage problems and cumulative coverage problem. Therefore, using conclusion of the continuity of coverage and cumulative cover problems can express the upper and lower bounds of the intermittent coverage problem.The third step to analyze and compute coverage area function is to analyze the parameters of ground target. In this paper, an assertion was given:any reasonable division mode of global or regional can correspond to a method of calculating coverage problem of constellation to ground. There are three kinds of the most natural way about global division. The first way is dividing global into a number of grid, the second way is dividing global into a number of longitude bands along meridian direction, the third way is dividing global into a number of latitude bands along the weft direction. Three global division methods correspond to the three coverage problem solving methods. The first division method corresponds to net-point method. The latter two division methods correspond to the longitude method and latitude method which is put forward in this paper.Through this three-step process, we can get the method for calculating coverage problem of constellation carrying arbitrary shape sensor and with arbitrary attitude to ground with arbitrary shape at any time request. Then the solving algorithm of coverage area function is obtained4. The analysis of the manifold structure spanned by the three elementsThe fourth major work of this paper is to analyze the manifold structure of decision space. The space spanned by the three elements is a three-dimensional manifold, and it is difficult to express directly. In this paper, according to the method of dividing any area into a set of latitude bounds, a numerical methods is used to change the three-dimensional manifold into a set of two-dimensional manifolds. Every two-dimensional manifold can be expressed by two graphs, which are called the upper and lower bounds of visual range longitude figure. The combination of all visual range longitude figure can construct this three-dimensional manifold. Due to visual longitude range graph stands for the manifold of decision space itself, its projection to ground target space is the coverage area function, and its projection to timeline is coverage time function.On the basis of the above work, a simulation platform of Earth-Mars transfer orbit design and optimization is build. Finally, the simulation results compared with the STK simulation results show that the results are correct, but also explain that the methods used in this paper are correct and effective.All the research work in this paper based on the three basic concepts:The first concept is make the problem mathematicization, and use the theory and conclusions of corresponding area of mathematics for them. The most direct embodiment of the idea is translate the coverage problem into several coverage functions, and among them, theory and conclusions of mathematical logic, set theory, measure theory, are applied to it. The second concept is unification, in this paper, through several cover function definition, most of the coverage problem is unified in a unified form, and calculated in a unified method. When calculating sensor cover range, a set of algorithms were put forward which can satisfy the sensor of any shape. The second concept is association. In this paper, several inner associations were found:coverage problem and time-varying characteristics calculation problem is a same manifold under different space projection; The constellation continuous coverage problem and cumulative coverage problem are the face of positive and negative of the same question. Coverage having largest interval problem is a coupling of continuous coverage problem and cumulative coverage problem.The follow-up work of this paper include the following points:1. The analysis of Cover time function. Most chapters of this article is the analysis of the state function, as well as decision space manifold structure of the coverage area function, however, involves less of coverage time function. But coverage time function can also make an analysis according to the framework presented in this paper.2. Establishment of coverage model function. In the idea of this article, in addition to the three coverage functions, another coverage function called coverage model function can also be built. The function use group theory to analyze the symmetry of coverage problem. At present, the theory is only partially completed, so it is not involved in the paper.3. The coverage problem under the orbit parameter with random error. The coverage problem under the orbit parameter with random error can also be incorporated into the theory of system established in this paper.