Quaternion Models and Algorithms for Solving the General Problem of Optimal Reorientation of Spacecraft Orbit and its Plane

Abstract

In this paper, the general problem of optimal reorientation of the spacecraft orbit and its plane is solved in a nonlinear formulation using the quaternion differential equation of the spacecraft orbit orientation and the Pontryagin maximum principle. It is necessary to minimize the functional, which is a weighted integral sum of time and energy spent on the reorientation process. In the process of reorientation of the orbit, its shape and size change. The control (the acceleration vector from the jet thrust) is limited in modulo. It is necessary to determine the optimal orientation of this vector in space and the law of optimal change of its module. A special case of the problem is the problem of optimal correction of the angular elements of the spacecraft orbit. This case has great importance in the mechanics of space flight. The paper provides concrete examples of numerical solutions to the general problem of optimal reorientation of the spacecraft orbit and its plane for the fast-response problem and the problem of minimum energy consumption. Graphs of changes in phase variables and optimal control are given. The analysis of the received solutions is carried out; their characteristic properties and regularities are revealed. In comparison with previous studies, we have obtained a solution in the case when the difference in the orientations of the initial and final spacecraft orbits equals to tens of degrees. At the same time, the combination of two methods for solving boundary value problems made it possible to increase the accuracy of the numerical solution of the boundary value problem from 0.002 to 1E-9 dimensionless units.