### On an approach to determining the orientation of the spacecraft orbit

#### Abstract

The paper is dedicated to a mathematical simulation of the motion of a spacecraft in an elliptical orbit. The acceleration vector is limited in modulo and orthogonal to the plane of spacecraft orbit during its motion. The spacecraft motion is described using a linear quaternion differential equation of the spacecraft orbit orientation. This equation has no singular points. Its approximate solution was found as a linear combination of basis functions. The cases are considered when the basis functions are polynomials or trigonometric functions. Unknown quaternion coefficients of this decomposition are found using the method of mean weighted residuals. The weight functions are Dirac delta functions. The analytical calculations are compared with the numerical solution of the Cauchy problem by the Runge-Kutta method of the 4th order of accuracy. Tables of the error in determining the orientation of the spacecraft orbit are given for the case when the initial position of the spacecraft orbit corresponds to the orientation of the orbit of one of the satellites of the GLONASS orbital grouping. It is shown that the decomposition of the desired solution by polynomials gives a smaller error than by trigonometric functions. We can determine the scalar part of desired quaternion with the smaller error than its vector part.

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