Two Spacecraft Pursuit-evasion Strategies On Low Earth Orbit And Numerical Solution

This thesis focuses on strategies and numerical methods of two spacecraft pursuitevasion problem on low Earth orbit based on zero-sum differential games theory.With the development of space technology, various satellites and spacecraft of different functions and applications were launched into outer space, and the number of spacecraft is increasing rapidly. As a result, aggressive behavior may occur between spacecraft inevitably. Therefore, pursuit-evasion problems for space vehicles have been increasingly studied. In a pursuit-evasion problem, the objective of the pursuing spacecraft is to approach the evading spacecraft, while the objective of the evading spacecraft is to avoid the pursuing spacecraft. Since the maneuverability has been greatly improved, it is possible for spacecraft to pursue and escape from each other. The pursuit-evasion problem of spacecraft is formulated as zero-sum differential games in this research. Different from the optimal control theory, zero-sum differential games theory investigates maximization and minimization of the payoff for two parties in the game, considering the influences of the strategy selections, rather than one-sided maximization or minimization in the optimal control theory.Zero-sum differential games theory has been well developed over the past sixty years. However, obtaining the solutions for differential game problems are still difficult in engineering applications. According to the existence conditions of saddle points for zero-sum differential games, the process of acquiring the solution of the game is actually to satisfy a boundary-value problem(BVP) of differential equations. The kinetic equations of spacecraft pursuit-evasion problems are the differential equations with the high dimensions, and it is rather difficult to solve the BVP due to the influences of orders and nonlinear terms. Therefore, the solution of pursuit-evasion problems for spacecraft is exceptionally challenging.This study focuses on strategies and numerical methods for spacecraft pursuit-evasion problems in low Earth orbit according to the latest development of the research for zerosum differential games theory and its numerical approaches. Main contents of this thesis are as follows:In Chapter 2, a mathematical model is introduced to describe the pursuit-evasion problem in which a virtual point is taken as the origin of the coordinate frame, and a numerical approach is presented by combining the improved non-dominated sorting genetic algorithm(NSGA-II) and the multiple shooting method. Since the coefficient matrix of the kinetic equations is nonlinear and time-dependent, and the low Earth orbit is an elliptical orbit of low eccentricity, the kinetic equations were expanded via the eccentricity ratio e of the orbit and the simplified pursuit-evasion model is established in the situation of the low eccentricity. Error analysis shows that the simplified model is reasonable in the sufficient precision. Based on the model, the saddle points solution is solved by the combination of NSGA-II and the multiple shooting method. The initial values for the multiple shooting method are provided by NSGA-II as a preprocessor.In Chapter 3, two numerical solutions: semi-direct piecewise nonlinear programming method(Semi-DPNLP) and Hybrid method are developed for the pursuit-evasion problem. Semi-DPNLP is based on the control parameterization method, and Hybrid method is formulated by Semi-DPNLP and the multiple shooting method, which integrates the convergence of Semi-DPNLP and the accuracy of the multiple shooting method.In addition, the equivalence between solutions of Semi-DPNLP and the original problem is proved according to the definition of the best reply strategy. It provides the theoretical foundation for the application of Semi-DPNLP and Hybrid method to solve the differential game problems. Furthermore, gradient formulas for the pursuit-evasion problem are derived in the use of Semi-DPNLP, and the solution to the problem is obtained by sequential quadratic programming method. Simulation results demonstrate that Semi-DPNLP and Hybrid method are both feasible for solving the pursuit-evasion problem.In Chapter 4, the feasibility of solving the multiple saddle points in differential game problems by the Semi-DPNLP and Hybrid method is discussed. A comparison principle is achieved by applying the viscosity solution method in Hamilton-Bellman systems, such that the uniqueness of the game value of the pursuit-evasion problem is proved. In other words, any saddle points corresponds to the same game value. This shows that the solutions obtained by the proposed Semi-DPNLP or Hybrid method must be the saddle point solutions, even when the pursuit-evasion problem may have several saddle points.In Chapter 5, the advantages of Semi-DPNLP and Hybrid method are presented,compared with the existing semi-direct collocation nonlinear programming method(SemiDCNLP) in the application of the spacecraft pursuit-evasion problem. Characteristics of the three methods are described separately. Theoretically speaking, these three methods are entirely different: Semi-DCNLP discretizes the state and control variables of the differential equations; Semi-DPNLP only discretizes the control variables; while Hybrid method is a combination of Semi-DPNLP and the multiple shooting method. Computational accuracy and computation time are compared by simulation examples to demonstrate the performance of the three methods for solving the pursuit-evasion problem. It is shown that the computation speed of Semi-DCNLP is the slowest; while that of SemiDPNLP is the fastest; Hybrid method has the highest accuracy among the three numerical methods. Therefore, the proposed Semi-DPNLP and Hybrid method are better solvers for this two spacecraft pursuit-evasion problem in low Earth orbit.