The use of quaternions in the optimal control problems of motion of the center of mass of a spacecraft in a Newtonian gravitational field: I

The problem of optimal control is considered for the motion of the center o f mass of a spacecraft in a central Newtonian gravitational field. For solv ing the problem, two variants of the equations of motion for the spacecraft center of mass are used, written in rotating coordinate systems. Both the variants have a quaternion variable among the phase variables. In the first variant this variable characterizes the orientation of an instantaneous or bit of the spacecraft and (simultaneously) the spacecraft location in this orbit, while in the second variant only the instantaneous orbit orientation is specified by it. The suggested equations are convenient in the respect that they allow the general three-dimensional problem of optimal control by the motion of the spacecraft center of mass to be considered as a composit ion of two interrelated problems. In the first variant these problems are ( 1) the problem of control of the shape and size of the spacecraft orbit and (2) the problem of control of the orientation of a spacecraft orbit and th e spacecraft location in this orbit. The second variant treats (1) the prob lem of control of the shape and size of the spacecraft orbit and the orbit location of the spacecraft and (2) the problem of control of the orientatio n of the spacecraft orbit. The use of quaternion variables makes this consi deration most efficient. The problem of optimal control is solved on the ba sis of the maximum principle. Several first integrals of the systems of equ ations of the boundary value problems of the maximum principle are found. T ransformations are suggested that reduce the dimensions of the systems of d ifferential equations of boundary value problems (without complicating them ). Geometrical interpretations are given to the transformations and first i ntegrals. The relation of the vectorial first integral of one of the derive d systems of equations (which is an analog of the well-known vectorial firs t integral of the studied problem of optimal control) with the found quater nion first integral is considered. In this paper, which is the first part o f the work, we consider the models of motion of the spacecraft center of ma ss that employ quaternion variables. The problem of optimal control by the motion of the spacecraft center of mass is investigated on the basis of the first variant of equations of motion. An example of a numerical solution o f the problem is given.