Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit

This paper considers a spacecraft pursuit-evasion problem taking place in low earth orbit. The problem is formulated as a zero-sum differential game in which there are two players, a pursuing spacecraft that attempts to minimize a payoff, and an evading spacecraft that attempts to maximize the same payoff. We introduce two associated optimal control problems and show that a saddle point for the differential game exists if and only if the two optimal control problems have the same optimal value. Then, on the basis of this result, we propose two computational methods for determining a saddle point solution: a semi-direct control parameterization method (SDCP method), which is based on a piecewise-constant control approximation scheme, and a hybrid method, which combines the new SDCP method with the multiple shooting method. Simulation results show that the proposed SDCP and hybrid methodsare superior to the semi-direct collocation nonlinear programming method (SDCNLP method), which is widely used to solve pursuit-evasion problems in the aerospace field

Minimum-Fuel Finite-Thrust Relative Orbit Maneuvers via Indirect Heuristic Method

Fuel-optimal space trajectories in the Euler–Hill frame represent a subject of great relevance in astrodynamics, in consideration of the related applications to formation flying and proximity maneuvers involving two or more spacecraft. This research is based upon employing a Hamiltonian approach to determining minimum-fuel trajectories of specified duration. The necessary conditions for optimality (that is, the Pontryagin minimum principle and the Euler–Lagrange equations) are derived for the problem at hand. A switching function is also defined, and it determines the optimal sequence and durations of thrust and coast arcs. The analytical nature of the adjoint variables conjugate to the dynamics equations leads to establishing useful properties of these trajectories, such as the maximum number of thrust arcs in a single orbital period and a remarkable symmetry property, which holds in the presence of certain boundary conditions. Furthermore, the necessary conditions allow translating the optimal control problem into a parameter optimization problem with a fairly small parameter set composed of the unknown initial values of the adjoint variables. A simple swarming algorithm is chosen among the different available heuristic techniques as the numerical solving algorithm, with the intent of finding the optimal values of the unknown parameters. Five examples illustrate the effectiveness and numerical accuracy of the indirect heuristic method applied to optimizing orbital maneuvers in the Euler–Hill frame

Optimal Interception of Optimally Evasive Spacecraft

This research addresses the problem of the optimal interception of an optimally evasive orbital target by a pursuing spacecraft or missile. The time for interception is to be minimized by the pursuing space vehicle and maximized by the evading target. This problem is modeled as a two-sided optimization problem, i.e. as a two-player zero-sum differential game. This work presents a recently developed method, termed "semi-direct collocation with nonlinear programming", devoted to the numerical solution of dynamic games. The method is based on the formal conversion of the two-sided optimization problem into a single-objective one, by employing the analytical necessary conditions for optimality related to one of the two players. An approximate, first attempt solution for the method is provided through the use of a genetic algorithm in the preprocessing phase. Three qualitatively different cases are considered. In the first example the pursuer and the evader are represented by two spacecraft orbiting Earth in two distinct orbits. The second and third case involve two missiles, and a missile that pursues an orbiting spacecraft, respectively. The numerical results achieved in this work testify to the robustness and effectiveness of the method also in solving large, complex, three-dimensional problems